Question

If the terms ${a_1}$ , ${a_2}$, ${a_3}$, ${a_4}$, ${a_5}$ are in AP with common difference $ \ne 0$, then find the value $\sum {i = {1^5}{a_i}} $ when ${a_3} = 2$.
1. $10$
2. $5$
3. $20$
4. None of these

Answer 1

Hint: In order to find the value of $\sum {i = {1^5}{a_i}} $, we need to know what the symbol is saying. This symbol stands for summation. It says the sum of the terms from 1 to 5, which is numerically written as ${a_1} + {a_2} + {a_3} + {a_4} + {a_5}$. Find the values, substitute them in the equation, solve and get the results.
Formula used:
${a_n} = {a_1} + \left( {n - 1} \right)d$

Complete step-by-step solution:
It is given that ${a_1}$ , ${a_2}$, ${a_3}$, ${a_4}$, ${a_5}$ are in AP and the value of ${a_3} = 2$.
And, we know that the term of an AP series is written as ${a_n} = {a_1} + \left( {n - 1} \right)d$.
So, substituting the value of ${a_3} = 2$ in the above formula and we get:
${a_3} = {a_1} + \left( {3 - 1} \right)d$
$ \Rightarrow 2 = {a_1} + 2d$
Subtracting both sides by $2d$:
$ \Rightarrow 2 - 2d = {a_1} + 2d - 2d$
$ \Rightarrow {a_1} = 2 - 2d$ ……(1)
Since, we obtained the value of ${a_1}$, so using this we can find the value of other terms, as we know that the next term of the series is the sum of the previous terms and the common difference(d), so from this statement the next terms are:
$ \Rightarrow {a_2} = {a_1} + d$
Substituting the value ${a_1} = 2 - 2d$, we get:
$ \Rightarrow {a_2} = 2 - 2d + d$
$ \Rightarrow {a_2} = 2 - d$ ….(2)
${a_3} = 2$ …..(3)
$ \Rightarrow {a_4} = {a_3} + d$
Substituting the value ${a_3} = 2$, above:
$ \Rightarrow {a_4} = 2 + d$ ……(4)
Similarly, for ${a_5}$:
$ \Rightarrow {a_5} = {a_4} + d$
Substituting the value of ${a_4} = 2 + d$ in the above equation, we get:
$ \Rightarrow {a_5} = 2 + d + d$
$ \Rightarrow {a_5} = 2 + 2d$ ……(5)

We are also given as $\sum {i = {1^5}{a_i}} $, which can be expanded as:
$\sum {i = {1^5}{a_i}} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5}$
Substituting the values of ${a_1},{a_2},{a_3},{a_4}{\text{ and }}{{\text{a}}_5}$ from the equation 1, 2, 3 ,4 and 5 in the above equation and we get:
$ \Rightarrow {a_1} + {a_2} + {a_3} + {a_4} + {a_5}$
$ \Rightarrow 2 - 2d + 2 - d + 2 + 2 + d + 2 + 2d$
On solving it further:
$ \Rightarrow 2 + 2 + 2 + 2 + 2$
$ = 10$
That implies
$\sum {i = {1^5}{a_i}} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} = 10$
Therefore, Option 1 is correct.

Note:
1). Arithmetic Progression is a series in which the two consecutive terms have a common difference, and it is followed for the whole series.
2). We can also find the summation directly with the formula ${a_1} + {a_5} = {a_2} + {a_4} = 2{a_3}$ as they are in AP and the value ${a_3}$ will be the Arithmetic mean. And we know that for ${a_1},{a_2},{a_3}$ in AP, ${a_2}$ will be the mean which is equal to half of the sum of the ${a_1},{a_3}$.