﻿ Arithmetic Sequences - Finding nth term - Free SAT practice: CrackSAT SAT Classes
Home     SAT practice     numbers & operations     sequence and series     arithmetic sequence - finding nth term

## SAT Arithmetic Progression Finding the nth term

### Question 1: Nth Term of an Arithmetic Sequence

What is 24th term of an arithmetic sequence if its 11th term is 29 and its 31st term is 169?

1. 140
2. 120
3. 91
4. 112
5. 76
Choice B. The 24th term of the sequence is 120.

SAT Math questions from Sequences and Series are relatively easy ones.

Use the formula given in the 'Hint / Formula' tab and try and solve the question. If you need further assistance, click on the tabs provided to the right of the hint / formula tab for a detailed step wise explanation to solve the question.

The nth term of an arithmetic sequence tn = t1 + (n - 1)d, where t1 is the first term of the arithmetic progression and 'd' is the common difference.

To find the 24th term of the arithmetic sequence, you will need the first term t1 and the common difference 'd'.

Data given: 11th term and 31st term of the same arithmetic sequence.

1. Express the 11th term and the 31st term in terms of the first term and common difference of the arithmetic progression.
2. Solve the two equations to get the first term and the common difference.
3. Use the values obtained to find the 24th term of the arithmetic sequence

### Step 1: Express the 11th and 31st term in terms of t1 and d

The 11th term of the sequence t11 = t1 + (11 - 1)*d

or t11 = 29 = t1 + 10d ..... eqn (1)

The 31st term of the sequence t31 = t1 + (31 - 1)d

or t31 = 169 = t1 + 30d ...... eqn (2)

Subtract eqn (1) from eqn (2)

t31 - t11 = 169 - 29 =140 = t1 + 30 d - (t1 + 10d)

or 140 = 20d

So, d = 7.

Substitute the value of d in eqn (1)

29 = t1 + 10*7

or t1 = 29 - 70 = -41.

### Step 2: Find the 24th term using t1 and d computed in step 1

t24 = t1 + (24 - 1)d

t24 = -41 + 23 * 7

t24 = -41 + 161

t24 = 120

### Alternatively: The difference between the 11th and 31st term is 20 common differences

The difference between any two consecutive terms of an arithmetic sequence is d. So, the difference between the 11th and the 31st term of an arithmetic sequence is 20 common differences.

t31 – t11 = 20d

169 – 29 = 140 = 20d

So, d= 7.

The difference between the 11th term and the 24th term is 13 common difference.

So, t24 = t11 + 13d

or t24 = 29 + 13*7

Hence, t24 = 29 + 91

t24 = 120

## Concept Overview : Arithmetic Progression

Get a quick overview on the basic concepts of Arithmetic sequence including formulae to find the nth term of an arithmetic sequence and the sum of n terms of an arithmetic sequence.

[an error occurred while processing the directive]