Chapter 8.2, Problem 82E

To determine

Whether sum of an arithmetic sequence can be found by only knowing its first term and last term.

Answer to Problem 82E

The statement is false.

Explanation of Solution

Given information:

The first term and the last term of a finite arithmetic sequence.

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Sum of an arithmetic finite sequence has the form

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

Here, n is number of terms, $a_1$ is the first term of the sequence, and $a_n$ is the last term of the sequence.

Now, from the formula it is clear that, the number of terms is a vital key to obtain the sum of a finite arithmetic sequence.

Therefore, it is impossible to calculate the sum of an arithmetic sequence can be found by only knowing its first term and last term.