Question

If the first, second and last terms of an A.P. are a, b and 2a respectively, the sum of the series is :
A) $ab/2(b - a)$
B) $3ab/2(b - a)$
C) $ab/(b - a)$
D) None of these

Answer 1

We have to find sum of the series using the formula ${S_n} = {S_n} = \dfrac{n}{2} \times (a + {a_n})$ and for that we need to find the value of ‘n’ (number of terms in the series ). Value of n can be calculated using the formula of last term which is ${a_n} = a + (n - 1)d$ and then the value of ‘n’ is to be put in the equation for finding the sum.

Complete step by step answer:
According to the question,
Given: 1st term ${a_1} = a$----(1)
2nd term ${a_2} = b$
Last term ${a_n} = 2a$-----(2)
Formula for ${a_n}$is $a + (n - 1)d$
(n= no. of terms in the series , d= difference between the two consecutive terms )
$d = {a_2} - {a_1}$
$d = b - a$
Here, we will put the values in this formula
${a_n} = a + (n - 1)d$
$2a = a + (n - 1)(b - a)$
$2a - a = bn - b - an + a$
$b = bn - an$
$n = b/(b - a)$------(3)
Now we will find sum of the series,
${S_n} = \dfrac{n}{2} \times (a + {a_n})$
Using the equation (1), (2) and (3 , we will be putting the values
${S_n} = \dfrac{b}{{2\left( {b - a} \right)}} \times (a + 2a)$
${S_n} = \dfrac{b}{{2(b - a)}} \times 3a$
${S_n} = \dfrac{{3ab}}{{2\left( {b - a} \right)}}$
Sum of the series is $\dfrac{{3ab}}{{2\left( {b - a} \right)}}$. So, the correct option is option (B).

Note: Whenever we need to find the sum of the series which are in arithmetic progression all the values such as d ( difference between the two consecutive terms ), ${a_n}$( last term of the series), n ( number of terms in the series ) and a ( first term of the series ) is to be calculated. The difference between two consecutive terms always remains constant in the series which are in arithmetic progression.