Question

The first term of an arithmetic progression is $2$ and the common difference is $4$. The sum of its $40$ terms will be:
(A) $3200$
(B) $1600$
(C) $200$
(D) $2800$

Answer 1

The given problem requires us to find the sum of an arithmetic progression. The first term and the common difference between two consecutive terms of the series is given to us in the question. For finding out the sum of an arithmetic progression, we need to know the first term, the common difference and the number of terms in the arithmetic progression. We substitute the values of a, n and d in the formula for calculating the sum of n terms of AP: $S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$.

Complete answer:
The first term of AP is given as $2$.
The common difference of an AP is given as $4$.
The number of terms in AP is $40$.
Now, we have to find the sum of this arithmetic progression.
Here, first term $= a = 2$.
Common difference $= d = 4$
Number of terms $= n = 40$
Now, we can find the sum of the given arithmetic progression using the formula $S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Hence, the sum of AP $= S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Substituting the values of a, d and n in the formula, we get,
$\Rightarrow S = \dfrac{{40}}{2}\left[ {2\left( 2 \right) + \left( {40 - 1} \right)\left( 4 \right)} \right]$
Opening the brackets and cancelling the common factors in numerator and denominator, we get,
$\Rightarrow S = 20\left[ {4 + 39 \times 4} \right]$
Simplifying the expression by doing the calculations, we get,
$\Rightarrow S = 20\left[ {4 + 156} \right]$
Adding like terms, we get,
$\Rightarrow S = 20 \times 160$
$\Rightarrow S = 3200$
So, the sum of the given $40$ terms of an arithmetic progression whose first term is $2$ and the common difference is given as $4$ is $3200$.
Hence, option (A) is the correct answer.

Note:
Arithmetic progression is a series where any two consecutive terms have the same difference between them. The common difference of an arithmetic series can be calculated by subtraction of any two consecutive terms of the series, if not given in the question itself. The sum of n terms of an arithmetic progression can be calculated if we know the first term, the number of terms and difference of the arithmetic series as: . We can also calculate the sum of terms in AP using the formula $S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ where l denotes the last term of arithmetic progression.