Question

The sum of how many terms of the A.P. $22,20,18,...$will be zero?

Answer 1

Hint: From the given terms of A.P, we can find the 1st term a and the common difference d. Then we can find the sum of the A.P. for n terms using the equation ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$. Now we can equate the sum to zero. Then we can find the number of terms by solving for n.

Complete step by step answer:

We have the A.P. $22,20,18,...$
From the A.P. the 1st term is 22. $ \Rightarrow a = 22$
The common difference can be obtained by taking the difference between two consecutive terms.
$d = {a_2} - {a_1}$
On substituting the values, we get,
$d = 20 - 22$
$ \Rightarrow d = - 2$
The sum of n terms of an A.P.is given by the equation, ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$.
On substituting the values of a and d, we get,
${S_n} = \dfrac{n}{2}\left( {2 \times 22 + \left( {n - 1} \right)( - 2)} \right)$
On further simplification, we get,
${S_n} = n\left( {22 - \left( {n - 1} \right)} \right)$
$ \Rightarrow {S_n} = n\left( {23 - n} \right)$
We need to find the number of terms that make the sum as 0. For that we equate the sum to zero and solve for n.
On equating the sum to zero, we get,
$n\left( {23 - n} \right) = 0$
$ \Rightarrow n = 0$or $23 - n = 0$
We cannot have the value 0 for n because number of terms in the given series cannot be equal to 0, so we reject $n = 0$.
$ \Rightarrow 23 - n = 0$
$ \Rightarrow n = 23$
Therefore, the sum of 23 terms of the A.P. will be zero.

Note: A.P. or arithmetic progression is the sequence of numbers which have a constant difference between any 2 consecutive numbers. This constant difference is called common difference. If the 1st term of an A.P. is a and the common difference is d, the nth term of the A.P. is given by the equation, ${a_n} = a + \left( {n - 1} \right)d$ and the sum of the n terms is given by ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$. For an A.P. the value of n will be a positive integer. It cannot have negative values, fractional values or decimals. So we reject the values which do not belong the set of natural numbers.