Number of integral terms in the expansion of \ 7 ^ ((1)/(2)) + 11 ^ ((1)/(6)) \ ^ 824 is equal to ______.

JEE Main 202430 JanuaryMorning Shift

Exam: JEE Main
Year: 2024
Date: 30 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2024 · 30 January · Morning Shift · Mathematics · Binomial Theorem

Number of integral terms in the expansion of \ 7 ^ ((1)/(2)) + 11 ^...

Number of integral terms in the expansion of {\left\{{7}^{\left(\dfrac{1}{2}\right)} + {11}^{\left(\dfrac{1}{6}\right)}\right\}}^{824} is equal to ______.

Correct answer

Step-by-step solution

We know that,

General term in expansion of ${\left(( 7{)}^{1 / 2} + ( 11{)}^{1 / 6}\right)}^{824} is given by,$ {t}_{r + 1} = Cr 824 \cdot 7\dfrac{824 - r}{2} \cdot 11\dfrac{r}{6} $For integral term,$ r $must be multiple of$ 6 $and should be less than$ 824 $So, the series will be$ r = 0, 6, 12, . . . . . 822 $Now, finding number of terms we get,$ 822 = 0 + \left(n - 1\right)6 $$ ⇒ 137 = \left(n - 1\right) $$ ⇒ n = 138 $Hence, total integral terms will be$ 138

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