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Arithmetic Progressions — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsArithmetic Progressions4 Marks
Question
Write the expression $a_n - a_k$  for the A.P. $a, a + d, a + 2d, ...$
Hence, find the common difference of the A.P. for which,
$11^{th}$​​​​​​​ term is $5$ and $13^{th}$​​​​​​​ term is $79$.
✓

Answer

We know,
$a_n=a+(n-1) d$
$\text { Let, }$
$n^{\text {th }} \text { term } a_n=a+(n-1) d$
$\Rightarrow a_n=a+n d-d$
$k^{\text {th }} \text { term, } a_k=a+(k-1) d$
$\Rightarrow a_k=a+k d-d$
Now,
$\Rightarrow a_n-a_k=(a+n d-d)-(a+k d-d)$
$\Rightarrow=a+n d-d-a-k d+d$
$\Rightarrow=n d-k d$
$\Rightarrow=d(n-k)$
Given, $11^{\text {th }}$ term, $a_{11}=5$
and $13^{\text {th }}$ term, $a_{13}=79$
We know, an $=a+(n-1) d$
then,
$11^{\text {th }} \text { term, } a_{11}=a+(11-1) d$
$\Rightarrow 5=a+10 d \ldots . .(i)$
$13^{\text {th }} \text { term, } a_{13}=a+(13-1) d$
$\Rightarrow 79=a+12 d . . . . . \text { (ii) }$
By subtituting eq. (i) from eq. (ii)
$\Rightarrow 79-5=a+12 d-(a+10 d)$
$\Rightarrow 74=a+12 d-a-10 d$
$\Rightarrow 74=2 d$
$\Rightarrow d=37$

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