Chapter 9.2, Problem 85E

To determine

the error in below calculation showing the sum of the first 50 odd integers.

  $\begin{array}{c}{S}_n=\frac{n}{2}\left(a_1+a_n\right)\\ =\frac{50}{2}\left(1+101\right)\\ =2550\end{array}$

Answer to Problem 85E

  $a_{50}=99$

Explanation of Solution

Concept Used:

The n^th term of an arithmetic sequence with first term $a_1$ and common difference d is given by:

  $a_n=a_1+\left(n-1\right)d$

The sum of first n terms of an arithmetic sequence with first term $a_1$ and last term $a_n$ is given by:

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

The error here is in the last term.

The sequence of odd integers is an arithmetic sequence with first term $a_1=1$ and $d=2$ .

Then, the 50^th odd integer is:

  $\begin{array}{c}{a}_{50}=a_1+\left(50-1\right)d\\ =1+49\left(2\right)\\ =1+98\\ =99\end{array}$

But in the question last term is taken to be 101 which is the error.

The correct sum of first 50 odd integers is:

  $\begin{array}{c}{S}_{50}=\frac{50}{2}\left(a_1+a_{50}\right)\\ =25\left(1+99\right)\\ =25\left(100\right)\\ =2500\end{array}$