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Question 1
Find a if the 17th and 18th terms of the expansion (2 + a)50 are equal.
We know that
General term of expansion (a + b)n is
Tr+1 = nCr an–r br
Finding 17th term
T17 = T16 + 1 of (2 + a)50
Putting r = 16, n = 50, a = 2 and b = a in (1)
T16 + 1 = 50C16 (2)50 – 16 . (a)16
T17 = 50C16 . (2)34 . a16
Finding 18th term
T18 = T17+1 of (2 + a)50
Putting r = 17, n = 50, a = 2 and b = a in (1)
T17+1 = 50C17 (2)50 – 17 . (a)17
T18 = 50C17 (2)33. (a)17
Now it is given that
17th term = 18th term
50C16 (2)34 . (a)16 = 50C17 (2)33 . a17
("50C16 " . 234)/("50C17 ." 233" " ) = 𝑎17/𝑎16
"50C16" /"50C17 " . (2)34 – 33 = a
"50C16" /"50C17 " × 21 = a
(50!/16!(50 −16)!)/(50!/17!(50−17)!) × 2 = a
50!/(16! × 34!) × (17! × 33!)/50! × 2 = a
(17 × 16! × 33!)/(16! × 34 × 33!) × 2 = a
17/34 × 2 = a
17/17 = a
a = 1
Hence, a = 1

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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