Number of integral terms in the expansion of \( \left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824} \) is equal to ______.
Correct Answer: 138
Solution and Explanation
Solution: The general term in the expansion is:
\[ t_{r+1} = \binom{824}{r} \left( \sqrt{7} z \right)^{824 - r} \left( \frac{1}{6 \sqrt{z}} \right)^r \]
which simplifies to \( z^{\frac{824 - r}{7} - \frac{r}{6}} \).
For an integral power, \( r \) must be a multiple of 6.
Thus, \( r = 0, 6, 12, \dots, 822 \), giving 138 integral terms.
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Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .