Question

In the expansion of $(5^{\frac{1}{2}}+7^{\frac{1}{8}}{)}^{1024}$, the number of

integral terms is

• A. 128
• B. 129
• C. 130
• D. 131

Answer 1

The correct option is B

129

Here n = 1024 = $2^{10}$, a power of 2, where as the

power of 7 is $\frac{1}{8}$ = $2^{-3}$

Now first term ${}^{1024}{C}_0$($5^{\frac{1}{2}}{)}^{1024}$ = $5^{512}$ (integer)

And after 8 terms, the $9^{th}$ term = ${}^{1024}{C}_8$($5^{\frac{1}{2}}{)}^{1016}$ ($7^{\frac{1}{8}}{)}^8$ = an integer

Again, ${17}^{th}$ term = ${}^{1024}{C}_{16}$($5^{\frac{1}{2}}{)}^{1008}$ ($7^{\frac{1}{8}}{)}^{16}$

= An integer.
Continuing like this, we get an A.P. 1, 9, 17,..........., 1025,

because ${1025}^{th}$ term = the last term in the expansion

= ${}^{1024}{C}_{1024}$($7^{\frac{1}{8}}{)}^{1024}$ = $7^{128}$ (an integer)

If n is the number of terms of above A.P. we have

1025 = $T_n$ = 1 + (n - 1)8 ⇒ n = 129.