Permutations and Combinations

Question

# How many terms of the A.P. -6, -5, .......... are needed to give the sum -25? Explain the double answer.

The given A.P. is: -6, -5, .............
Here, a = -6,
Let -25 be the sum of n terms

∴              or
or        or
or                       or                   n(n - 25) = -100
or
or

Here, the common difference is positive.

∴ The A.P. starts from negative terms and its terms are increasing.

∴ All the terms after 13th term are positive.

These positive terms from 14th term to 20th term when added to negative terms from 6th term to 12th term, they cancel out each other and the sum remains same.

Hence, the sum of first five terms is same as the sum of first 20 terms.