The number of integral terms in the expansion of (3^(1/2)+5^(1/4))⁶⁸⁰ is equal to ___________.

Solution

Maths sir · 4 steps

Write the general term, ask when both powers are whole numbers, then count those values of r.

1. General term. The (r+1)th term is a binomial coefficient times a power of 3 times a power of 5. The binomial coefficient is always an integer, so only the two powers decide whether the term is an integer.

   Tᵣ₊₁ = C(680, r) 3^((680-r)/2) 5^(r/4), r = 0, 1, …, 680

2. Conditions on r. 5^(r/4) is an integer only when r is a multiple of 4. Then r is even, and since 680 is even, (680-r)/2 is also an integer. So the only condition is that 4 divides r.
3. Count the values. The allowed values are r = 0, 4, 8, …, 680. There are 680/4 = 170 steps of 4, plus the starting value r = 0, so 170 + 1 = 171 values.
4. So. The expansion has 171 integral terms.

The idea to keep. For integral terms, make every fractional exponent in Tᵣ₊₁ an integer, combine the conditions into one LCM condition on r, and count from r = 0 up to n.

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