)r =7Cr⋅x7n−nr−5r⋅2r For positive power of x, 7n−nr−5r>0 ⇒7n>r(n+5) ⇒r<
7n
n+5
As r represent term of binomial expression so r is always integer. Given that sum of coefficient is 939. When r=0, sum of coefficient =7C0⋅20=1 when r=1, sum of coefficient =7C0⋅20+7C1⋅21=1+14=15 when r=2, sum of coefficient =7C0⋅20+7C1⋅21+7C2⋅22 =1+14+84 =99 when r=3, sum of coefficient =7C0⋅20+7C1⋅21+7C2⋅22+7C3⋅23 =1+14+84+280 =379 when r=4, sum of coefficient =7C0⋅20+7C1⋅21+7C2⋅22+7C3⋅23+7C4⋅24 =1+14+84+280+560 =939 To get value of r=4, value of
7n
n+5
should be between 4 and 5 . ∴4<
7n
n+5
<5 ⇒4n+20<7n<5n+25 ∴4n+20<7n ⇒3n>20 ⇒n>
20
3
⇒n>6.66 and 7n<5n+25 ⇒2n<25 ⇒n<12.5 ∴6.66<n<12.5 ∴ Possible integer values of n=7,8,9,10,11,12 ∴ Sum of values of n=7+8+9+10+11+12=57