Question

Determine the 288th term of the series a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f

• A. u
• B. x
• C. v
• D. w
• E. y

Hint:

For series with increasing repetitions, use triangular numbers to find the group containing the desired term.

Answer 1

The Correct Option is A

Step 1: Pattern: a appears 1 time, b appears 2 times, c appears 3 times, d appears 4 times, etc. So the $n$th letter appears $n$ times.
Step 2: Cumulative sum up to letter $m$: $S_m = 1 + 2 + 3 + \dots + m = \frac{m(m+1)}{2}$.
Step 3: Find smallest $m$ such that $S_m \ge 288$. $\frac{m(m+1)}{2} \ge 288 \implies m(m+1) \ge 576$. $24 \times 25 = 600 \ge 576$, so $m = 24$ gives $S_{24} = 300$. $S_{23} = \frac{23 \times 24}{2} = 276$.
Step 4: Since $S_{23} = 276$, the 277th term is the start of letter u (24th letter). 288 - 276 = 12th occurrence of u.
Step 5: So the 288th term is u.
Step 6: Final Answer: u.