## Formula's for the nth term

Here I intend to look at what is known as the **n ^{th}**

**term**sequence. In doing so we will look at some examples and find the

**n**which is known in mathematical terms by a

^{th }term**"u"**followed by a subscript(small)

_{"}_{n}_{"}_{}.

So what exactly is the connection( the pattern) between the **"n"** and the **"u _{n}".**

If we take a closer look at the **u _{n }**

_{}we can see that it goes up in

**two's (2's).**

**(Position) U 1 2 3 4 5**

** ( Term) U _{n }3 5 7 9 11**

** n x 2 2 4 6 8 10**

**3 + 2 = 5**

**5 + 2 = 7**

**7 + 2 = 9**

**9 + 2 = 11**

Therefore we can write down that **u _{n }= 2n**, in other words we are doubling the number.

Looking again we may need to **adjust that 2n slightly**;

**2n means 2 x n therefore,**

**2 x 1 = 2**

**2 x 2 = 4**

**2 x 3 = 6**

**2 x 4 = 8**

**2 x 5 = 10**

These are the numbers we have got after multiplying**n x 2**, you will have noticed that they are **1 **less in each case than the numbers in the **u _{n}**

_{}bracket, therefore we need to

**add on 1**in the formula.

Our formula now looks like this **u _{n }= 2n + 1 (where the u_{n }is the n^{th }term)**

**2 x n = ? + 1 = u _{n}**

**2 x 1 = 2 + 1 = 3 **

**2 x 2 = 4 + 1 = 5**

**2 x 3 = 6 + 1 = 7**

**2 x 4 = 8 + 1 = 9**

**2 x 5 = 10 + 1 = 11**

**Answer:- u _{n }= 2n + 1**

**This is the type of question for the n ^{th }^{}term, could be asked in an exam paper;**

**Find the n ^{th} term for the sequence; 2, 6, 10, 14, 18 ... (etc)?**

**To begin with we need to change this sequence into a table!**

**The "n" is the position and the “u _{n}” is the term.**

**The position simply represents the number of digits in a sequence, in the above there are "5" [ 2(which is 1), 6(which is 2), 10(which is 3), 14(which is 4), 18( finally this is number 5 )]**

**(Position) U 1 2 3 4 5**

_{}

** ( Term) U _{n }2 6 10 14 18**

** n x 4 4 8 12 16 20**

**So what exactly is the connection (the pattern ) between the "n" and the "u _{n}", in this question?**

**If we take a closer look at the "u _{n}" we can see that it is going up in four's (4's).**

** 2 + 4 = 6**

** 6 + 4 = 10**

**10 + 4 = 14**

**14 + 4 = 18 **

**Therefore we can write down that** **u _{n }= 4n (which means 4 x n)**

**4 x 1 = 4**

**4 x 2 = 8**

**4 x 3 = 12**

**4 x 4 = 16**

**4 x 5 = 20**

**These are the numbers we have got after multiplying n x 4, you will have noticed that there are 2 more in each case than the numbers in the u _{n }term, therefore we need to subtract 2 in the formula.**

**Our formula now looks like this; u _{n }= 4n - 2 (where the u_{n}_{}is the n^{th }term)**

**4 x n = ? - 2 = u _{n }**

**4 x 1 = 4 – 2 = 2**

**4 x 2 = 8 – 2 = 6**

**4 x 3 = 12– 2 = 10**

**4 x 4 = 16– 2 = 14**

**4 x 5 = 20– 2 = 18**

**Answer:- u _{n }= 4n - 2**

**In this example we need to find the n ^{th }term for this sequence;**

**Find the n ^{th} term for the sequence; 3, 6, 11, 18, 27 ... (etc)?**

**To begin with we need to change this sequence into a table!**

**This example is quite similar to the one above!!!!**

**LOOK CAREFULLY:-**

**The "n" is the position and the “u _{n}“ is the term**

**Again there are five numbers and they are holding the first five POSITIONS**

**The position simply represents the number of digits in a sequence, in the above there are "5" [ 3(which is 1), 6(which is 2), 11(which is 3), 18(which is 4), 27( finally this is number 5 )]**

**(Position) U 1 2 3 4 5**

** ( Term) U**_{n }**3 6 11 18 27**

**n ^{2}=(n x n) 1 4 9 16 25 **

**So what exactly is the connection (the pattern ) between the "n" and the "u _{n}", in this question? **

**Is there a set pattern?**

**Are they going up in 2's or 4's? --- NO they are not!**

** 3 + 3 = 6**

** 6 + 5 = 11**

**11 + 7 = 18**

**18 + 9 = 27**

In a case like this we would square (**n ^{2}**) the numbers and see what that brings us too.

**Therefore we can write down that u _{n }= n^{2} (which means n x n)**

**1 x 1 = 1**

**2 x 2 = 4**

**3 x 3 = 9**

**4 x 4 = 16**

**5 x 5 = 25**

**These are the numbers we have got after multiplying n x n, you will have noticed that they are 2 less in each case than the numbers in the u _{n }term, therefore we need to add 2 in the formula.**

**Our formula now looks like this; u _{n }= n^{2} + 2 ( where the u_{n }is the n^{th }term)**

**n x n = ? + 2 = u _{n}**

**1 x 1 = 1 + 2 = 3**

**2 x 2 = 4 + 2 = 6**

**3 x 3 = 9 + 2 = 11**

**4 x 4 = 16 + 2 = 18**

**5 x 5 = 25 + 2 = 27**

**Answer:- u _{n }= n^{2} + 2 **

**Another favourite question in exams is as follows;**

**The n ^{th }term of a sequence is represented by this formula:- u_{n }= 3n + 2**

**We are asked to;**

**(i) Find the first 4 terms**

**(ii) To find the 49^{th} term **

**( i ) Find the first 4 terms**

**Our formula is:- 3n + 2 = u _{n }**

**Therefore our 1 ^{st} term is u_{1} = 3 x 1 = 3 + 2 = 5**

** 2 ^{nd} term u_{2 }= 3 x 2 = 6 + 2 = 8**

** 3 ^{rd} term u_{3 }= 3 x 3 = 9 + 2 = 11**

** 4 ^{th} term u_{4 }= 3 x 4 = 12 + 2 = 14**

**Our first four terms are; u _{1} , u_{2} , u_{3 }, and u_{4 }**

**They end up as 5, 8, 11, and 14**

**\ First 4 terms are: 5, 8, 11, 14. (Answer:-)**

**( ii ) We have been asked to find the 49 ^{th} term. **

**49 ^{th} term: This will be known as u_{49}**

**As before in part ( i ) the formula is; u _{n }= 3n + 2 **

**Therefore we will replace the “ _{n}” with the 49**

** u _{n }= 3n + 2**

**u _{49 }= (3 x 49) = 147 + 2 = 149**

**(Answer) u _{49 }= 149**

## Comments

**artina** on March 17, 2018:

what is the nth term of 12,23,36,51,68

**john** on April 19, 2015:

find the nth term of 2n+5. find the frist term,second term and common difference

**amberb** on September 24, 2014:

HELP ASAP the first 5 terms and the 100th term for the sequence 0.5n-0.5

**thalene** on March 01, 2014:

the sum of the 17 term of an arithmetic sequence is 2414 and the last term is 256. find the first term

**wow** on February 23, 2014:

Not bad

**pooja** on January 08, 2014:

This website is boaring

**Avicii** on January 07, 2014:

Does anybody on this goddamn blog answer questions!!!??

**Telamon** on November 03, 2013:

what is the first 5 terms and the 100th term for the sequence whose rule is 0.5n - 0.5

**Rues** on November 01, 2013:

24,54,103,....

**becky** on April 17, 2013:

how would I solve for the nth term:

1+2=(2x3)/2

1+2+3=(3X4)/2

1+2+3+4=(4x5)/2

**Naseem Choudhury** on January 10, 2013:

what is the term of 25-4n

**kola** on December 18, 2012:

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20=210

**tom** on September 23, 2012:

what's the nth term for 10, 40, 90, 160, 250, 360, 490...

**ray** on September 05, 2012:

what's 5 ways to find the 77th term?

**Cherry** on July 09, 2012:

N term of these number 13,21,37,69,133. Please help

**Kiki** on June 14, 2012:

Given the progression 7, 12, 17 and 22. What is the expression for the nth term of the progression?

**Howard** on June 13, 2012:

what is the pattern of these numbers?

a)2,3,6,85,68

thank you

**timmi2000** on June 11, 2012:

wondering if anyone could help me with these questions, and help me work it out as i do not seem to get it.

1) find the 8th term of each of these sequences

a) 2,5,8,11

b) 3,7,11,15

c) 2,10,50,250

d) 2,6,18,54

2) find the sum of the first 10 terms of the following series

a)3,8,13,18

b)1,3,9,27

3) the first term of an arithmetic series is 1. the common difference is 6 find the 10th term of the series, the sum of the first N terms is 7400 find N

**met** on May 31, 2012:

what are the next two numbers 4, 5, 14, 15, 40, 41, _, _, ?

**luiza** on May 26, 2012:

Please help me

look at the equation n+3=12. Use iot to work out the value of n-3

**Kyle** on May 24, 2012:

Which statement describes how to find the nth term in the sequence?

12, 29, 46, 63, 80, ...

A.add 5 times the term number and 12

B.subtract 17 times the term number from 12

C.subtract 5 from 17 times the term number

D.add 5 and 17 times the term number

**Adam Saleh** on May 23, 2012:

The nth term of a sequence is n^2-6n-4, find the sum of 3rd and 4th terms

**chris** on May 14, 2012:

An arithmetic sequence has first term a=6 and common difference d=12. How many terms of this sequence must be added to get 5040?

**Pavan Kumar** on May 09, 2012:

24 24 28 46 94 what is the next sequence

**ragu** on May 07, 2012:

In the sequence, 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 when we add any three numbers we have to get the answer 30

**yaya** on April 26, 2012:

hi am year 7 what is the 5 terms and the 10th term of 2n+2

**fawzy** on April 26, 2012:

plz define in detail what do we do with sequences not having a common difference E.G 0..-1..-8..-27..-64.

And also do you happen to know any formula like

Tn= A+d(n-1)+ 1/2[(n-1)(n-2)c] which claims to work on everything. If Yes than please explain the formula.

**Helmi** on April 01, 2012:

how to solve this

U 1 2 3 4 5

Un 1 4 10 22 46

**Lisa** on March 25, 2012:

trying to find out the correct sequence for if n+1/3

**Subhasish Raha** on March 20, 2012:

what is the missing number in the following series

9,19,40,___,146?

**suruthi** on March 19, 2012:

if the sum of 4 consecutive terms in a gp is 60 and if the t1,18,t4 are in ap find the terms

ans(4,8,16,18)

**Kalyanakrishnan** on March 17, 2012:

give solution to this sum

sum of the series 2+4(3*3)+6(5*5)+.........+22(21*21)

**Jimmy** on March 15, 2012:

nth term of sequence the sequence is 4, 12, 24

This is what i gett

4 12 24 40

\ / \ / \ /

8 12 16

\ / \ /

4 4

But I dont know how to make a formula from this help

**SURUTHI** on March 14, 2012:

HI,

pleasssssssse solve this problem ,i am not getting the SOLUTION anywhere in the net.

"the sum of the first n terms of an ap are in the ratio (3n +8):(7n + 15), find the ratio of the 3 rd terms"

{ans=23:50}.

**SURUTHI** on March 14, 2012:

HI,

pleasssssssse solve this problem ,i am not getting the solution anywhere in the net.

"the sum of the first n terms of an ap are in the ratio (3n +8):(7n + 15), find the ratio of the 3 rd terms" ans 23:50.

**CHRISTY** on March 12, 2012:

nth term of sequence ?7, ?21, 3?7.....

**williamovely@yahoo.com** on March 09, 2012:

a series is s

uch that the sum of the first nth therms is 2n^2 minus 3n.

1.how to Prove that the series is arithmatic

2.find the value of the 18th term

**Fazla** on March 07, 2012:

Maa''m i can't understand general term.. I don't knw why.. If u can help me in dis.. it will be a great help..

-2,4,-8,16.. can u help me 2 find da general term of dis set??

**dmndmz** on March 04, 2012:

Greetings ;)

What is the formula for the sequence 16, 29, 48, 52, 54 and 5 ?

**DIANA** on March 04, 2012:

WHAT IS THE FORMULER FOR THE SEQUENCE; 22,18,14,10?

**paul** on February 29, 2012:

in an AP the fifth term is 8 and the nineth term is 14. fin the first term.

**brittany** on February 26, 2012:

Can Anyone Find What's The 6th Term For 5 , 7 , 11 ....

**john** on February 26, 2012:

terms: 1,2,3,4,5,...n

values: 1,2,4,8,16,...

please find me the formula?

**waleedrehman** on February 25, 2012:

plz find the indicated term of the following,1,3/2,5/4,7/8,...a7.

**alicia** on February 24, 2012:

what is the 75th term 8, 14, 20, 26, 32

**Jim** on February 23, 2012:

The nth term of a sequence is the l.c.m of integers from 1 to n. Which term of the sequence is the first one that's divisible by 100?

**Charlott3_Fay3** on February 22, 2012:

What is 2-3G and 7-1/2n??? I'm really confused and it's annoying me please answer tonight xxxx

**anonymous** on February 21, 2012:

given the general form an^2+bn+c

find a

pls help with the above

**M** on February 18, 2012:

What is the Nth Term to: -1, 7, 18, 29, 42, 57

and 2, 9, 18, 29, 42, 57?

Please help!

**F** on February 07, 2012:

5 x 4 x 3 ... -5 x -2 x - 4 Write out the correct form of the sequence

**sita** on February 04, 2012:

how to find the 40 term of a sequance with term to term rule add 5and the 1st term of 6

**Anne David** on February 03, 2012:

and the tenth term of 0.4,0.04,0.004,,, plz give me a formula for this

**Anne David** on February 03, 2012:

what is the formula of sixth term of 3/4,3,12...

**t.elia** on February 03, 2012:

how bout sixth term of 3/4,3,12...

**Geralyn.** on February 03, 2012:

the radiator of the truck contains 12 gallons of water. We draw off 3 gallons and replace it with alcohol, then we draw off 3 gallons of the mixture then replaced it by alcohol. until 8 drawings and replacements have been dine. How much alcohol is in the final mixture?

**Ronak** on February 03, 2012:

what will be the next term to the series 2,3,4,9,16... and how it will be???.....

**shubham** on February 02, 2012:

iam confuse between nth and n terms of A.P

PLEASE HELP T.ELIA

**akanksha raj** on February 01, 2012:

what will be the next term for this series 2,3,4,7,13,12,12,23,20,17,33,28,? please help.

**Vin** on February 01, 2012:

What's The Next Three Terms of the Arithmetic Sequence?

6,12,18,24,__,___,___?

Help Please :D

**Filip** on January 30, 2012:

-1 , -2 , -1 , 2 , 7 , 14 what's the nth term please?

**Simon** on January 29, 2012:

I'm stuck on this question so could you help me out?

All terms in a geometric sequence are positive integers, and the first three ter4ms are n, n+3, and 2n+6. What is the fourth term of this sequence?

**arihant** on January 29, 2012:

what is the nth term of 1+(1+2)+(1+2+3).....to n terms

**joyce** on January 28, 2012:

what would be the nth term of 30,28,26,24,22..

**joyce** on January 28, 2012:

what would be the nth term of 30,28,26,24,22... and what would be the nth term of 40,37,34,31,28...

**grace** on January 28, 2012:

what would be the nth term of 40,38,36,34,32...

**rohit choudahry** on January 25, 2012:

0,2,4,6,8,12,12,20,16,?

answer is 12.

can u explain the trick behind this?

**eivlys** on January 20, 2012:

Find the next term of the progression 2 4 6 14 23

**Carl** on January 19, 2012:

Hi would you be able to help me with the general term (tn) I guess, I doesn't know the word in english... well here are the numbers, 2,7,15,26,40 thanks :)

**Kieran** on January 17, 2012:

What is the nth term for this sequence (or similar ones): 2, 7, 15, 26. It goes up like this: +5, +8, +9.

**miley** on January 11, 2012:

how do you find the equation and 50th term of 1,4,9,16,25,36,48

**Marc** on January 11, 2012:

Find the next term of the progression 24,12,8,.....

**t.elia (author)** from Northern Ireland on January 10, 2012:

Simba,

What are you working on multiplication?? Please clarify.

Telia

**t.elia (author)** from Northern Ireland on January 09, 2012:

MAHI,

Arithematic Progression:-

A sequence of numbers. Let's start with a number: a1.

Now add a number d, (for "difference").

We get a1 + d and the first 2 terms in our sequence are:

a1, a1 + d

For the next term, let's add d to that last term and we have a1 + 2d.

Our sequence is now:

a1, a1 + d, a1 + 2d

We continue this process and the resulting set of numbers is called an arithmetic progression (AP) or arithmetic sequence

Good Luck Telia

Sorry posted the comment and wasn't signed in.

**MAHI** on January 09, 2012:

WHAT IS AP?

**ogleoo** on January 08, 2012:

i keep forgetting that n means term number!!!

**ogleoo** on January 06, 2012:

thank you so much

**t.elia (author)** from Northern Ireland on January 06, 2012:

Ogleoo,

48 - 4n

48 - (4 x 1)= 48 - 4 = 44

48 - (4 x 2)= 48 - 8 = 40

48 - (4 x 3)= 48 - 12 = 36 etc

I hope you find this easy to follow

Good Luck

**student** on December 12, 2011:

hello plaese help me find Recurrence Relations

t(n)=t(3n/5)+t(2n/5)+n

**t.elia (author)** from Northern Ireland on December 10, 2011:

Hi Paul, this link you provided isn't working.

**Paul** on December 10, 2011:

My homework sheet says that i have to write a rule in terms of n, i don't know what that means, the website for my homework is http://docs.google.com/viewer?a=v&q=cache:ap2t.../maths/docs/building_number_patterns.doc+building+investigative+skills&hl=en&gl=uk&pid=bl&srcid=ADGEESgdc2qKUJbnfZK9K6L22BqBGd27GIaVEhDj6m_QE5xBedMtgqj5QD7hfd1vx0armiQYpFn4mPs83vidpckpAmTdbMgP_2PR7u30s46xICSkUbnCcSq1TaTrXxr-ywh0FKdy2oaH&sig=AHIEtbScDxgftte_31i6WwekjrGnqCDRgA, i also don't get question 4

**t.elia (author)** from Northern Ireland on November 20, 2011:

Hi Jamie

Your answer is 3n + 9

**Jamie** on November 20, 2011:

Hi, can you help me.

If i have these 4 numbers, 12 15 18 21, what's the sequente to get the nth term?

**shreyansh** on November 19, 2011:

The value of (2/3)(3/4)(4/5).....(1-1/n)=

**t.elia (author)** from Northern Ireland on November 01, 2011:

Oscar,

Answer is 3n + 4

3 x 1 = 3 + 4 = 7

3 x 2 = 6 + 4 = 10

3 x 3 = 9 + 4 = 13 etc

Good Luck & hope this helps

**oscar** on October 31, 2011:

can you solv please 7,10,13,16

**bubble 543** on October 22, 2011:

what would the answer be for the nth term for a sequence is n-2/ n squared

write the first three terms of the sequence

**t.elia (author)** from Northern Ireland on October 16, 2011:

LIV

26-4n

26 - 4 x N

26 - 4 x (1) = 22

26 - 4 x (2) = 18

26 - 4 x (3) = 14 etc

Hope this helps

Good Luck

**liv** on October 16, 2011:

26-4n?

**Mahmoud Shahoud** on October 10, 2011:

Hi , You're smart , but all comments don't help me .

can you help me .

I have this sequence .

1 1 1 1 1 1 1 1 1 2

I want to calculate the 11st element and 12nd element

I want Math Equations .

I have a product gave me the answer but I want the way .

11st term = 11

12nd term = 56

please help me .

**t.elia (author)** from Northern Ireland on October 06, 2011:

DAVID

53, 56, 61, 68

first difference is 3, 5, 7

second difference is 2

Tn = a + (n - 1) d

The Sum of the n terms is written as Sn and we know that the nth term = Tn = Sn - (Sn-1)

Tn = 53 + 3 + 5 + 7

Tn = 53 + n/2[ 2*3 + (n - 1) 2]

Tn = 53 + n/2 x 6 + 2(n - 1) (Cancel down)

Tn = 53 + 3n + n^2 - n

Tn = n^2 + 2n + 53 (Answer)

Good Luck

**t.elia (author)** from Northern Ireland on October 06, 2011:

DAVID

50, 48, 46, 44

Tn = a + (n - 1) d

a = 50

d = 48

48 - 50 = -2

Tn = a + (n - 1) d

Tn = 50 + (n - 1) -2

Tn = 50 - 2n + 2

Tn = 52 - 2n (Answer)

Good Luck

**t.elia (author)** from Northern Ireland on October 06, 2011:

DOM & DAVID,

you have both asked for the same problem here;

3, 8, 15, 24, 35

First difference is 5, 7, 9, 11

Second difference is 2

The Sum of the n terms is written as Sn and we know that the nth term = Tn = Sn - (Sn-1)

Tn = a + (n - 1) d

Tn = n/2 [ 2a + (n - 1) d ]

Tn = n/2 [ 2*3 + (n - 1) 2]

Tn = n/2 x 6 + n/2 x (n - 1)2 (cancel down)

Tn = 3n + n^2 - n

Tn = n^2 + 2n (Answer)

Good Luck

**David** on October 06, 2011:

Excuse me, would you help me with 50,48,46,44 and 3,8,15,24,35 and 53,56,61,68 please, thanks.

**dom** on October 06, 2011:

3,8,15,24,35 what is the rule for that? im stuck !

**t.elia (author)** from Northern Ireland on October 05, 2011:

RUSS & ME,

Here is help with the sequence for your formula;

1 , 8 , 9 , 12

1^2, 2^3, 3^2 , (2*3*2) = 12

So the first number is squared = 1

...the second number is cubed = 8

...the third number is squared = 9

...the fourth is the powers multiplied (2*3*2) = 12

The fifth number will be 5^2 = 25

The sixth number will be 6^3 = 216

The seventh number is 7^2 = 49

The eighth will be the powers multiplied (2*3*2) = 12

and so on ......

Hope this helps

Good Luck

**Me** on October 05, 2011:

WHAT IS THE FORMULA FOR tHIS SEQUENCE PLEAAAASE! 1 8 9 12

**Russ** on October 05, 2011:

Can somebody help me with the sequence of 1 8 9 12 please...are there 2 sequences? What follows after 12? Much appreciated!

**Bex** on October 04, 2011:

What's the nth term for: -1 +1 -1 +1 -1.

Doing my GCSEs and stuck on homework.

**renee** on October 04, 2011:

what are the first 5 nth term of 3n+6

**saf** on October 04, 2011:

ok i know this is random but if you start with 2 and double each time then what is the number if the nth term is n+3

then 105-5n

2n-0.5v

thanx