Question

Is the statement true or false: doubling the number of terms in an arithmetic series, but keeping the first term and common difference the same, will double the sum?

Answer 1

False. The sum will not be the same unless the common difference is `0`.

Explanation:

Given an arithmetic sequence `a_1, a_2, ..., a_n` with `n` terms, we can calculate the sum of the sequence as

`sum_(k=1)^na_k = n(a_1+a_n)/2`

Now let's see what happens to the sum if we keep `a_1` and `d` the same, but change `n` to `2n`:

`sum_(k=1)^(2n)a_k = 2n(a_1+a_(2n))/2 = n(a_1+a_(2n))`

Is this double the first sum? Well, if we double the first sum, we get

`2sum_(k=1)^na_k = 2n(a_1+a_n)/2 = n(a_1+a_n)`

So, double the first is only equal to the second if

`n(a_1+a_(2n)) = n(a_1+a_n)`

which is true if and only if `a_(2n) = a_n`

But later terms in the sequence are only equal to earlier terms if the common difference is `0`. Thus, the statement is false in general, and true only if the common difference is `0`.