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# The sum of first 8 terms of an A.P is 100 and ${S_{19}} = 551$. Find $a$ and $d$.

Last updated date: 18th Sep 2024
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Hint:
The sum of n terms of AP is the sum(addition) of first n terms of the arithmetic sequence. It is equal to n divided by 2 times the sum of twice the first term – ‘a’ and the product of the difference between second and first term-‘d’ also known as common difference, and (n-1), where n is the number of terms to be added. Use this relation to find two linear equations of two variables and solve them to get the values of a and d.

Complete step by step solution:
It is given that,
The sum of the first 8 terms of the A.P is = 100
The sum of the first 19 terms is = 551
Let The first term of the A.P be $a$and the common difference be $d$.
The sum of n terms of AP is the sum(addition) of first n terms of the arithmetic sequence. It is equal to n divided by 2 times the sum of twice the first term – ‘a’ and the product of the difference between second and first term- ‘d’ also known as common difference, and (n-1), where n is the number of terms to be added.
Sum of n terms of AP = $\dfrac{n}{2}[2a + (n - 1)d]$
We have been given that,
${S_8} = 100$
$\Rightarrow \dfrac{8}{2}[2a + (8 - 1)d] = 100$
$\Rightarrow 4[2a + 7d] = 100$
$\Rightarrow 2a + 7d = 25$……… (i)
And, ${S_{19}} = 551$
$\Rightarrow \dfrac{{19}}{2}[2a + (19 - 1)d] = 551$
$\Rightarrow 19[a + 9d] = 551$
$\Rightarrow a + 9d = 29$…………… (ii)
We will solve eq. (i) and eq. (ii) to get the values of a and d.
First, multiply eq. (ii) with 2. We will get,
$2a + 18d = 58$…………… (iii)
Now. Subtract eq. (i) from eq. (iii). So, We get,
$2a + 18d - 2a - 7d = 58 - 25$
$\Rightarrow 11d = 33$
$\Rightarrow d = 3$
Now, substitute $d = 3$ in eq. (i). We will get the value of a.
$2a + 7 \times 3 = 25$
$\Rightarrow 2a + 21 = 25$
$\Rightarrow 2a = 25 - 21$
$\Rightarrow 2a = 4$
$\Rightarrow a = 2$

Hence, the required values are $a=2$ and $d=3$.

Note:
Once you find the value of a and d, you can form the A.P as follows.
The AP will be $a, a + d, a + 2d, a + 3d,............, a + (n - 1)d$.
So, the AP is: 2, 5, 8, 11, 14 and so on.