Chapter 12.2, Problem 45AYU

In Problems 39-56, find each sum.

$73+78+83+88+\cdots +558$

To determine

To find: The sum of the given sequence.

Answer to Problem 45AYU

$30,919$

Explanation of Solution

Given:

$73+78+83+88+\dots +558$

Formula Used:

If $a_1,a_n$ and $d$ are known, then $n$^th term can be found by the formula,

$a_n=a_1+\left(n-1\right)d$   (Formula 1)

Sum of the first $n$ terms of an arithmetic sequence.

$S_n=\frac{n}{2}\left[a_1+a_n\right]$   (Formula 2)

Calculation:

To find $n$:

From the given sequence, $a_1=\text{ 73},a_n=\text{ 558 and }d=\text{ 5}$.

$\text{558 }=\text{ 73 }+\left(n-1\right)\text{5 }=\text{ 73 }+\text{ 5}n-\text{ 5 }=\text{ 5}n\text{ + 68}$   ( By Formula 1)

$\Rightarrow n=\text{ 98}$

To find the sum $S_{98}$:

$S_{98}=\frac{98}{2}\left[73+558\right]=49\times 631=30,919$   (By Formula 2)