Question

what is the average of all the multiple of 9 lying between 120 and 280?​

Answer 1

❏ ForMuLaS Used

if in an A.P. series a be the first term and d be the common difference then ,

(1) The n'th term is given by the formula .

$\sf\longrightarrow\boxed{ a_n=a+(n-1)d }$

(2)Sum of n number of terms ,

$\sf\longrightarrow\boxed{ S_n=\frac{n[2a+(n-1)d]}{2} }$

Now

Average formula,

$\sf\longrightarrow\boxed{ AVG.=\frac{Sum\: of\: terms}{total \:number\: of \:terms} }$

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❚ Let's Solve The Problem ❚

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❏ Solution

Q) what is the average of all the multiple of 9 lying between 120 and 280?

ans)

The multiples of 9 that lying between 120 to 280 are,

126,135,144,.......,279

By studying the above A.P series,

first term (a)=126

common difference (d)=9

let, 279 is the n'th term of this A.P. series.

$\therefore a_n=a+(n-1)d$

$\sf\longrightarrow 279=126+(n-1)9$

$\sf\longrightarrow 279-126=(n-1)9$

$\sf\longrightarrow n=\frac{153}{9}+1$

$\sf\longrightarrow n=17+1$

$\sf\longrightarrow \boxed{n=18}$

∴ sum of 18 terms is ,

$\bf\longrightarrow \:S_{18}=\frac{\cancel{18}[2a+(18-1)d]}{\cancel2}$

$\sf\longrightarrow \:S_{18}=9[2(126)+17(9)]$

$\sf\longrightarrow \:S_{18}=9[405]$

$\sf\longrightarrow\boxed{ \:S_{18}=3645}$

Now, the average of all the multiple of 9 lying between 120 and 280 is,

$\sf\longrightarrow S_{avg}=\frac{S_{18}}{18}$

$\sf\longrightarrow S_{avg}=\frac{\cancel{3645}}{\cancel{18}}$

$\sf\longrightarrow \boxed{S_{avg}=202.5}$

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$\underline{ \huge\mathfrak{hope \: this \: helps \: you}}$

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