Chapter 12.2, Problem 35AYU

In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.

$14thtermis-1;18thtermis-9$

To determine

To find: The $1^{st}$ term, common difference, recursive formula and formula for $n$th term.

Answer to Problem 35AYU

1. $a_1=25, d=-2$

2. $a_n=a_{n-1}-2$

3. $a_n=27-2n$

Explanation of Solution

Given:

${14}^{th}$ term is $-1$ and ${18}^{\text{th}}$ term is $-9$.

Formula used:

Formula for $n^{th}$ term:

$a_n=a_1+\left(n-1\right)d$   Eq (1)

$a_n=a_{n-1}+d$   Eq (2)

Calculation:

From Eq (1) and given data,

$a_{14}=a_1+13d=-1$

$a_{18}=a_1+17d=-9$

Solving above two equations,

$d=-2$; $a_1=25$

From Eq(2), we get the recursive formula,

$a_n=a_{n-1}-2$

The formula for $n^{th}$ term is

$a_n=a_1+\left(n-1\right)d=25+\left(n-1\right)\left(-2\right)=-2n+27$