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Question

If A1,A2,.....A31 are arithmetic means inserted between the numbers a and b, then find the value of (b+A51bA51)(A1+aA1a)

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Solution

We know that

tn=a+(n1)d

b=a+(n1)d

d=ban+1

We have

A1=a+d=a+(ban+1)

A2=a+2d=a+2(ban+1)

An=a+nd=a+n(ban+1)


A1=a+d=a+(ba51+1)=a+(ba52)=b+51a52

A51=a+d=a+51(ba51+1)=a+51(ba52)=a+51b52

Now, (b+A51bA51)(A1+aA1a)

(b+a+51b52ba+51b52)(b+51a52+ab+51a52a)
On simplifying we get

103b+abab+103aba

103b+ab103aba

102b102aba=102

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