Question

The sum of seven consecutive odd integers is 133.What is the least number

Answer 1

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$\mathcal{\blue{\underline{TYPE OF PROBLEM:}}}$

(•) TO FIND SMALLER OR LARGER NO. WHEN AVERAGE OF 'n' CONSECUTIVE ODD NO.S IS KNOWN

$\mathcal{\red{\underline{APPROACH TO PROBLEM:}}}$

Let the odd No. s be :
x+1, x+3, x+5, ........ ,x+(2n+1)

Sum of these observations:
= nx +(1 + 3+ ………(2n+1))
= nx+ n²
(°•° Sum of odd No.s ,
1+3+ …+(2n-1) = n²)

Here,
Smaller No. = x+1
Larger No. = x+(2n-1)

$Average A = \frac{Sum Of Observations}{No. Of Observations}$

=> $A= \frac{nx+n^{2}}{n}$
=> nA = n(x+n)
=> A= (x+n)

=> Smallest No. x+1 = A-n+1
& Largest No. x+(2n-1) = An+(2n-1) = n(A+2)-1

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$\mathcal{\underline{\pink{SOLUTION:}}}$

Given,
No. of consecutive odd No.s "n" = 7
Sum of these observations = 133

First find Average of these observations
$A=\frac{133}{7}$

=>$A= 19$

Now, Using Above Formula,

Least No. = A-n+1= 19-7+1 = 13

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$\mathcal{\huge{\pink{Hope it helps}}}$