Question

insert 10 arithmetic mean between -5 and 17 and prove that their sum is 10 times the arithmetic mean between -5 and 17​

Answer 1

SOLUTION

GIVEN

Insert 10 arithmetic mean between -5 and 17

TO PROVE

Their sum is 10 times the arithmetic mean between -5 and 17

EVALUATION

First we have to Insert 10 arithmetic mean between -5 and 17

So this is an Arithmetic progression

Number of terms in the progression = 12

First term = a = - 5

Last term = l = 12th term = 17

Let Common Difference = d

Now 12th term = 17

$\implies \: \sf{a + (12 - 1)d = 17}$

$\implies \: \sf{ - 5 +11d = 17}$

$\implies \: \sf{ 11d =22}$

$\implies \: \sf{ d = 2}$

Now the 10 arithmetic mean between -5 and 17 are

-3, 1, 3, 5, 7, 9, 11, 13, 15

So their sum =

$\displaystyle \sf{ S= \frac{10}{2} \big( - 3 + 15\big) }$

$= \sf{5 \times 12}$

$\sf{ = 60}$

Again the arithmetic mean between - 5 & 17

$\displaystyle \sf{ M = \frac{ - 5 + 17}{2} }$

$\displaystyle \sf{ = \frac{ 12}{2} }$

$\displaystyle \sf{ = 6 }$

Therefore S = 10M

Hence the sum of 10 arithmetic mean between -5 and 17 is 10 times the arithmetic mean between -5 and 17

Hence proved

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