Let α > 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term β x - α , β ∈ N . Then α is equal to _____ .

Question

Let &#945; &#62; 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term &#946; x - &#945; , &#946; &#8712; N . Then &#945; is equal to _____ .

Quantrex Academy · Accepted Answer

Correct answer: 0

General term of the binomial expansion of x 2 3 + 2 x 3 30 is T r + 1 = C r 30 x 2 / 3 30 - r 2 x 3 r &#8658; T r + 1 = C r 30 &#183; 2 r &#183; x 60 - 11 r 3 So, for term of the form &#946; x - &#945; , we have 60 - 11 r 3 &#60; 0 &#8658; 11 r &#62; 60 &#8658; r &#62; 60 11 &#8658; r = 6 So, T 7 = C 6 30 &#183; 2 6 x - 2 We have also observed &#946; = C 6 30 2 6 is a natural number. &#8756; &#945; = 2

Let α > 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term β x - α , β ∈ N . Then α is equal to _____ .

Correct answer

Step-by-step solution

More from Binomial Theorem

Let &#945; &#62; 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term &#946; x - &#945; , &#946; &#8712; N . Then &#945; is equal to _____ .

Correct answer

Step-by-step solution

More from Binomial Theorem

Let α > 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term β x - α , β ∈ N . Then α is equal to _____ .