Question

$\huge \mathfrak \color{green}{question} :$
The sum of the first 8 terms of A.P. is 100 and the sum of its first 19 terms 551. Find the First term and the common difference of the A.P. ​

Answer 1

Answer:

.let a be the first term and d be the common difference

A/q

(8/2)(2a+(8-1)d)=100

=2a+7d=25. (1)

and (19/2)(2a+(19-1)d)=551

=2a+18d=58. (2)

subtracting equations (1) and (2),

11d =33

d=3

so, a=(25-21)/2=2

so, AP is 2,5,8,11,14,..

Answer 2

AnsWeR :

$\large \boxed{\sf a = 2} \ \ \ and \ \ \ \ \ \boxed{d=3}$

Solution :

Given, S8=100 and S19 = 551

$\large \color{red} \boxed{\sf Sn = \frac{n}{2}[2a+(n-1)d]}$

$\star S8=\frac{n}{2}(2a+7d) = 100 \\ \implies 2a+7d=25 ....(i)$

$\\ \\ \star S19=\frac{n}{2}(2a+18d) = 551 \\ \implies a+9d=29 ....(ii)$

$\colorbox{pink}{\sf solving \: (i)\: and \: (ii) \: simultaneously \: , we \: get \:}$

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Solving for a and d

$a + 9d=29...(ii)\\ \\ \implies a = 29-9d ....(iii) \\ \\ \sf Put \: the \: value \: of \: a \: in \: (i) \\ \\ 2a+7d=25 ....(i) \\ \\ 2(29-9d)+7d = 25 \\ \\ \implies 58 -18d + 7d = 25 \\ \\ -11d=-33 \\ \\ \implies \boxed{d=3}$

Now , put the value of d in (i)

⟹a + 9d=29...(i)

⟹a+9(3)=29

⟹a+27=29

$\boxed{\sf a=2}$

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$\large \colorbox{lightgreen}{a=2} \ \ \ \ \ \ \ \ \colorbox{lightgreen}{d=3}$