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Arithmetic Progressions — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsArithmetic Progressions5 Marks
Question
Let there be an A.P. with first term ' $a$ ', common difference ' $d$ '. If $a_n$ denotes in $n^{\text {th }}$ term and $S_n$ the sum of first $n$ terms, find. $n$ and $S_n$, if $a=5, d=3$ and $a_n=50$.
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Answer

Here, we have an A.P. whose $n ^{\text {th }}$ term $\left(a_n\right)$, first term (a) and common difference ( $d$ ) are given, We need to find the number of terms $(n)$ and the sum of first $n$ terms $\left(S_n\right)$.
Here,
First term (a) = 25
Last term ($a_n$) = 50
Common difference (d) = 3
So here we will find the value of n using the formula, $an = a + (n - 1)d$
So, substituting the values in the above mentioned formula
$50 = 5 + (n - 1)3$
$50 = 5 + 3n - 3$
$50 = 2 + 3n$
$3n = 50 - 2$
Furhter simplifying for n,
$3n = 48$
$\text{n}=\frac{48}{3}$
$n = 16$
Now, here we can find the sum of the n terms of the given A.P., using the formula,
$\text{S}_\text{n}=\Big(\frac{\text{n}}{2}\Big)(\text{a}+\text{l})$
Where, a = the first term
l = the last term
So, for the givne A.P, on substituting the values in the formula for the sum of n terms of an A.P., we get,
$\text{S}_{16}=\Big(\frac{16}{2}\Big)[5+50]$
$= 8(55)$
$= 440$
Therefore, for the given A.P. n = 16 and $S_n = 440$

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