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SAT Math : How to find the nth term of an arithmetic sequence

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Example Question #31 : Sequences

An arithmetic sequence begins as follows: 14, 27, 40...

What is the first four-digit integer in the sequence?

Possible Answers:

1,000

1,006

1,008

1,002

1,004

Correct answer:

1,002

Explanation:

Given the first two terms $a_{1}$ and $a_{2}$ , the common difference is equal to the difference:

$d = a_{2} - a_{1}$

Setting $a_{1} = 14$ , $a_{2} =27$ :

d = 27 - 14 = 13

The th term of an arithmetic sequence $a_{n}$ can be found by way of the formula

$a_{n} = a_{1}+ (n-1) d$

Since we are looking for the first four-digit whole number - equivalently, the first number greater than or equal to 1,000:

$a_{1}+ (n-1) d \ge 1,000$

Setting $a_{1} = 14$ and d = 13 and solving for :

$14 + (n-1) 13 \ge 1,000$

$14 + 13n-13 \ge 1,000$

$13n+1 \ge 1,000$

$13n+1 - 1 \ge 1,000 - 1$

$13 n \ge 999$

$13 n \div 13 \ge 999 \div 13$

$n \ge 76\frac{11}{13}$

Therefore, the 77th term, or $a_{77}$ , is the first element in the sequence greater than 1,000. Substituting n = 77 , $a_{1} = 14$ , and d = 13 in the rule and evaluating:

$a_{77}=a_{1}+ (n-1) d$

= 14+ (77-1) 13

$= 14+76 \cdot 13$

= 14+988

=1,002 ,

the correct choice.

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SAT Math : How to find the nth term of an arithmetic sequence

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