Chapter 11.5, Problem 21WE

To determine

To calculate:

Sum of first $20$ positive integers ending in $3$ .

Answer to Problem 21WE

Sum of first $20$ positive integers ending in $3$ is $1960$ .

Explanation of Solution

Give information:

First $20$ positive integers ending in $3$

Calculation:

The sequence of positive integers ending in $3$ is arithmetic.

  $a_n=3+\left(n-1\right)10$

First $20$ positive integers ending in $3$ ,

  $a_{20}=3+\left(20-1\right)10=193$

To find sum of the $20$ terms in this sequence, “fold” the sequence is half by pairing the first term with the last term (i.e. $3$ with $193$ ), the second term with the second to last term, etc.

Altogether there will be $10$ such pairs and he sum of each pair is $196$ . Thus add up $196$ ten times.

  $\begin{array}{l}3+193=196\\ 196.10=1960\end{array}$

Therefore, Sum of first $20$ positive integers ending in $3$ is $1960$ .