Question

Find the sum of the AP: $-26,-24,-22,...$ to 36 terms.
(a) 324
(b) 314
(c) 389
(d) 349

Answer 1

Hint: Calculate the common difference of the given AP by subtracting two consecutive terms. To calculate the ${{n}^{th}}$ term of the AP, use the formula ${{a}_{n}}=a+\left( n-1 \right)d$, where ‘a’ is the first term of the AP, ‘n’ is the number of terms in the AP, ‘d’ is the common difference of AP and ${{a}_{n}}$ is the ${{n}^{th}}$ term of the AP. To calculate the sum of ‘n’ terms, use the formula $\dfrac{n}{2}\left( a+{{a}_{n}} \right)$ or $\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$.

Complete step-by-step answer:

We have to calculate the sum of 36 terms of the AP $-26,-24,-22,...$.
We will calculate the ${{n}^{th}}$ term of the AP using the formula ${{a}_{n}}=a+\left( n-1 \right)d$, where ‘a’ is the first term of the AP, ‘n’ is the number of terms in the AP, ‘d’ is the common difference of AP and ${{a}_{n}}$ is the ${{n}^{th}}$ term of the AP.
We observe that we have $a=-26,n=36$. We will calculate the common difference ‘d’ of the given AP by subtracting any two consecutive terms.
Thus, we have $d=-24-\left( -26 \right)=-24+26=2$.
Substituting these values in the given expression, we have ${{a}_{36}}=\left( -26 \right)+\left( 36-1 \right)2$.
Simplifying the above expression, we have ${{a}_{36}}=-26+70=44$.
We will now calculate the sum of 36 terms of this AP. To do so, we will use the formula ${{S}_{n}}=\dfrac{n}{2}\left( a+{{a}_{n}} \right)$.
Substituting $a=-26,{{a}_{36}}=44,n=36$ in the above formula, we have ${{S}_{36}}=\dfrac{36}{2}\left( 44-26 \right)=18\left( 18 \right)=324$.
Hence, the sum of 36 terms of the given AP is 324, which is option (a).

Note: We can also solve this question by using the formula ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ for calculating the sum of the first ‘n’ terms of the given AP. One must keep in mind that we can calculate the common difference by subtracting any two consecutive terms of the AP.