Question

The number of integral terms in the expansion of $ \left( 5^{\frac{1}{2}} + 7^{\frac{1}{8}} \right)^{1016} $ is:

• A. 130
• B. 128
• C. 127
• D. 129

Hint:

For binomial expansions involving fractional exponents, ensure that the exponents of the terms are integers for the terms to be integral. This can be done by ensuring that the powers of the terms satisfy the divisibility conditions.

Answer 1

The Correct Option is B

Step 1: General Form of the Expansion The given expression is of the form \( (a + b)^n \), where: \[ a = 5^{\frac{1}{2}}, \quad b = 7^{\frac{1}{8}}, \quad n = 1016. \] The general term in the binomial expansion of \( (a + b)^n \) is: \[ T_r = \binom{n}{r} a^{n-r} b^r. \] Substituting \( a = 5^{\frac{1}{2}} \) and \( b = 7^{\frac{1}{8}} \), we get the general term: \[ T_r = \binom{1016}{r} \left( 5^{\frac{1}{2}} \right)^{1016-r} \left( 7^{\frac{1}{8}} \right)^r = \binom{1016}{r} \cdot 5^{\frac{1016 - r}{2}} \cdot 7^{\frac{r}{8}}. \] Thus, the general term is: \[ T_r = \binom{1016}{r} \cdot 5^{\frac{1016 - r}{2}} \cdot 7^{\frac{r}{8}}. \]

Step 2: Identifying the Conditions for Integral Terms For the term \( T_r \) to be an integer, both \( 5^{\frac{1016 - r}{2}} \) and \( 7^{\frac{r}{8}} \) should be integers.
This means that the exponents of 5 and 7 must be integers. For \( 5^{\frac{1016 - r}{2}} \) to be an integer, \( \frac{1016 - r}{2} \) must be an integer, implying that \( 1016 - r \) must be even.
Therefore, \( r \) must be even. For \( 7^{\frac{r}{8}} \) to be an integer, \( \frac{r}{8} \) must be an integer, implying that \( r \) must be a multiple of 8.

Step 3: Finding the Range of \( r \) Since \( r \) must be an even number and a multiple of 8, \( r \) must be a multiple of 8.
The possible values of \( r \) are given by the set of multiples of 8, i.e., \( r = 0, 8, 16, \dots, 1016 \). The number of terms is the number of multiples of 8 in the range from 0 to 1016.
The multiples of 8 in this range are \( 0, 8, 16, \dots, 1016 \), which form an arithmetic progression with the first term 0, the common difference 8, and the last term 1016. The number of terms in this progression is: \[ \frac{1016 - 0}{8} + 1 = 128. \] Thus, there are 128 integral terms in the expansion.