The number of integral terms in the expansion of ( 512 + 718 )¹⁰¹⁶ is:

Solution

Maths sir · 3 steps

Write the general term and ask when both powers become whole numbers. It all comes down to r.

1. General term. The (r+1)th term is Tᵣ₊₁ = C(1016, r) 5^((1016-r)/2) 7^(r/8), for r = 0, 1, …, 1016. The binomial coefficient is always an integer, so only the two powers matter.
2. Conditions on r. 7^(r/8) is an integer only when r is a multiple of 8. 5^((1016-r)/2) is an integer only when 1016 - r is even. Any multiple of 8 is even, and 1016 is even, so the second condition holds automatically.

   r = 0, 8, 16, …, 1016 ⇒ (1016 - 0)/8 + 1 = 127 + 1 = 128

3. So. There are 128 values of r, so there are 128 integral terms, which is option (b).

The idea to keep. For integral terms, make every exponent in Tᵣ₊₁ a whole number. Find the LCM condition on r, then count terms in that AP from 0 to n, and do not forget r = 0.

Answer from the NTA final answer key. Solution by PrepWiser.