Term From The End

Question

In the expansion of (a + b)^{17}, a problem asks which numbered term counted from the beginning coincides with the 5th term counted from the very end of the expansion.

RankGuru · Accepted Answer

Counting terms from the end is converted to counting from the start using the rule that the rth term from the end is the (n - r + 2)th term from the beginning in an expansion of (a + b)^n, a useful JEE Advanced symmetry observation. The expansion of (a + b)^{17} has n + 1 = 18 terms in total. The 5th term from the end is the (18 - 5 + 1)th = 14th term from the beginning, since position from start plus position from end equals total terms plus one. Hence the 5th term from the end is the 14th term from the beginning. Option 13th term miscounts by one, forgetting the plus one in the indexing. Option 12th term subtracts incorrectly. Option 15th term overshoots the position. The relation position-from-start + position-from-end = total + 1 is the controlling identity. Plausibility check: the 14th term from the start and the 5th term from the end together give 14 + 5 = 19 = 18 + 1, exactly matching the total-terms-plus-one rule, which confirms consistency.

Term From The End

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