Chapter 12.2, Problem 33AYU

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.

$15thtermis0;40thtermis-50$

To determine

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

Answer to Problem 33AYU

1. $a_1=28, d=-2$

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

3. $a_n=30-2n$

Explanation of Solution

Given:

${15}^{th}$ term is 0 and ${40}^{\text{th}}$ term is $-50$.

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,

$a_{15}=a_1+14d=0$

$a_{40}=a_1+39d=-50$

Solving above two equations,

$d=-2$; $a_1=28$

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

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

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

$a_n=a_1+\left(n-1\right)d=28+\left(n-1\right)\left(-2\right)=30-2n$