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Question

If the coefficients of three consecutive terms in the expansion of 1+an are in the ratio 1:7:42 then n is divisible by


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Solution

Step 1: Use the general term of binomial expansion to find coefficient of three consecutive terms

The binomial expansion formula is

a+bn=i=0nCinan-ibi

Let the three consecutive term be

r-1thterm,rthtermandr+1thterm

The coefficient of r-1thterm is

Cr-1-1=Cr-2nn

The coefficient of rthterm is

Cr-1=Cr-1nn

The coefficient of r+1thterm is

Cr+1-1=Crnn

Step 2: Form equation using given ratio

The given ratio of the coefficient is 1:7:42

Taking the r-1thterm and rthterm we have

Cr-2nCr-1n=17n!r-2!n-r+2!n!r-1!n-r+1!=17r-1!n-r+1!r-2!n-r+2!=17r-1n-r+2=177r-7=n-r+2n-8r+9=0...................(1)

Taking the rthterm and r+1thterm we have

Cr-1nCrn=742n!r-1!n-r+1!n!r!n-r!=16r!n-r!r-1!n-r+1!=16rn-r+1=166r=n-r+1n-7r+1=0...................(2)

Step 3: Solve the obtained equations

Subtracting (1) and (2) we have

r-8=0r=8

Putting the value of r in equation (1) we have

n-8×8+9=0n=55

Hence the value of nis 55


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