In the expansion of (512+718)1024, the number of integral terms is

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Question

In the expansion of

(5^{\frac{1}{2}} + 7^{\frac{1}{8}})^{1024}

, the number of integral terms is

Options:

A . 128

B . 129

C . 130

D . 131

Answer: Option B
:
B
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^{t h}

term =

^{1024} C_{8}

(

5^{\frac{1}{2}})^{1016}

(

7^{\frac{1}{8}})^{8}

= an integer
Again,

17^{t h}

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^{t h}

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.

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