Question

How many terms are needed in the AP - 24, 20, 16... to make the sum 72?

Answer 1

Hint: In this particular type of question we need to find the common difference and the first term and equate it with the formula of sum of n terms of an AP which will help us to find the number of terms required to give a sum of 72.

Complete Step-by-Step solution:
Given A.P is 24,20,16..
So, the first term, a=24
The common difference, d=20−24=−4.
We know that the sum of first n terms of A.P is given by = $\dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] = \dfrac{n}{2}\left[ {48 - \left( {n - 1} \right)4} \right]$
But the sum is given 72.
Therefore ,
$
\dfrac{n}{2}\left[ {48 - \left( {n - 1} \right)4} \right] = 72 \\
\Rightarrow 24n - n\left( {n - 1} \right)2 = 72 \\
\Rightarrow 2{n^2} - 26n + 72 = 0 \\
\Rightarrow {n^2} - 13n + 36 = 0 \\
\Rightarrow n = 4{\text{ or 9}} \\
$
Hence 4 terms or 9 terms add up to 72.

Note: It is important for us to know the reason for getting two values of n. Since the common difference is a negative value the AP becomes negative after a period of time thus it gives the same value of the sum when added after n = 4 and n= 9.