Question

How do you find the first three terms of the arithmetic series ${a_1} = 17, {a_n} = 197$ and ${S_n} = 2247$?

Answer 1

First of all, mention the formulas for the ${n^{th}}$ term of A. P. and the sum of n terms of A. P. Thus, two equations with two unknown variables are there, which can be solved using substitution.

Complete step-by-step solution:
We are given that we are required to find the first three terms of the arithmetic series ${a_1} = 17,{a_n} = 197$ and ${S_n} = 2247$.
Since, we know that the formula for the ${n^{th}}$ term of an A. P. is given by the following
expression:-
$\Rightarrow {a_n} = {a_1} + (n - 1)d$, where n is the number of term and d is the common
difference.
According to the given data, we have ${a_1} = 17,{a_n} = 197$. Putting these in the above
mentioned formula, we will then obtain the following equation with us:-
$\Rightarrow 197 = 17 + (n - 1)d$
Taking 17 from addition in the right hand side to subtraction in the left hand side, we will then obtain the following equation with us:-
$\Rightarrow 180 = (n - 1)d$ ………………(1)
Since, we know that the formula for the sum of n terms of an A. P. is given by the following
expression:-
$\Rightarrow {S_n} = \dfrac{n}{2}\left[ {{a_1} + {a_n}} \right]$, where n is the number of term and d
is the common difference.
Putting the given values in the above equation, we will then obtain the following equation with us:-
$\Rightarrow 2247 = \dfrac{n}{2}\left[ {17 + 197} \right]$
Simplifying the calculations in the right hand side, we will then obtain the following equation with us:-
$\Rightarrow 2247 = 107n$
Taking 107 from multiplication in the right hand side to division in the left hand side, we will then obtain the following equation with us:-
$\Rightarrow \dfrac{{2247}}{{107}} = n$
Simplifying the calculations in the left hand side, we will then obtain the following equation with us:-
$\Rightarrow n = 21$
Putting this in equation number 1, we will then get the following:-
$\Rightarrow 180 = 20d$
Taking 20 from multiplication in the right hand side to division in the left hand side, we will then
obtain the following equation with us:-
$\Rightarrow d = 9$
Now, the second term of A. P. will be ${a_2} = {a_1} + d = 17 + 9 = 26$, third term will be ${a_3} = {a_1} + 2d = 17 + 18 = 35$.
Thus, the first three terms are 17, 26 and 35.

Note: The students must note that here, we had two equations with two unknown variables in it, therefore, we could solve them to find the values of both the unknown variable. If you notice carefully, we did not actually require the end process, but we had to find it, so that we could put it in the first equation to have the value of d which was actually required.