Find the next two terms of the series: A, C, F, J, ?, ? (A). O (B). U (C). R (D). V

Convert each letter to its position in the English alphabet: $ A 1,\; C 3,\; F 6,\; J 10. $ Look at the numeric sequence of positions: \[ 1,\;3,\;6,\;10,\; \] Compute the differences: \[ 3-1=2, 6-3=3, 10-6=4. \] The successive increments are $2,3,4, $ — that is, the increments increase by 1 each time. Therefore the next increments should be $5$ and $6$. So the next positions are \[...

Find the next two terms of the series: A,... - Solved

Question 70

Logical Reasoning Pattern Series and Sequences Easy

Find the next two terms of the series: $$ \text{ A, C, F, J, ?, ? } $$ $$ (A). O $$ $$ (B). U $$ $$ (C). R $$ $$ (D). V $$

(A) (A) and (B) respectively

(B) (B) and (A) respectively

(C) (C) and (D) respectively

(D) (D) and (C) respectively

✓ Correct Answer: Option A Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

Convert each letter to its position in the English alphabet:
$ A\to 1,\; C\to 3,\; F\to 6,\; J\to 10. $
Look at the numeric sequence of positions: \[ 1,\;3,\;6,\;10,\;\dots \] Compute the differences: \[ 3-1=2,\quad 6-3=3,\quad 10-6=4. \] The successive increments are $2,3,4,\dots$ — that is, the increments increase by 1 each time.
Therefore the next increments should be $5$ and $6$. So the next positions are \[ 10+5=15,\qquad 15+6=21. \]
Convert these positions back to letters: \[ 15 \mapsto O,\qquad 21 \mapsto U. \] So the next two terms are $ \boxed{O,\;U} $.
(Alternative observation) The positions $1,3,6,10,15,21,\dots$ are the triangular numbers $T_1,T_2,T_3,\dots$ with $T_n=\dfrac{n(n+1)}{2}$. The $n$-th term picks the letter at alphabet position $T_n$.

Final answer: The next two letters are O and U.

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