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Arithmetic Progressions — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsArithmetic Progressions4 Marks
Question
Let there be an A.P. with first term $'a'$, common difference $'d'$. If $a_n$​​​​​​​ denotes in $n^{th}​​​​​​​$​​​​​​​ term and $S_n​​​​​​​$​​​​​​​ the sum of first $n$ terms, find.
d, if $a = 3, n = 8$ and $S_n = 192$.
✓

Answer

Here, we have an A.P. whose first term $(a)$, Sum of first n terms $(S_n)$ and the number of terms $(n)$ are given. We need to find common difference $(d)$.
Here,
First term $(a) = 3$
Sum of n terms $(S_n) = 192$
Number of terms $(n) = 8$
So here we will find the value of n using the formula, $a_n = a + (n - 1)d$
So. to find the common difference of this A.P., we use the following formula for the sum of n terms of an A.P.
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
Where; $a$ = first term for the given A.P.
$d$ = common difference of the given A.P.
$n$ = number of terms
So, using the formula for $n = 8$, we get,
$\text{S}_8=\frac{8}{2}[2(3)+(8-1)(\text{d})]$
$192 = 4[6 + (7)(d)]$
$192 = 24 + 28d$
$28d = 192 - 24$
Furhter solving for d,
$\text{d}=\frac{168}{28}$
$d = 6$
Therefore, the common difference of the given A.P. is $d = 6.$

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