The sum of series represented as 1/(1*5) + 1/(5*9) + 1/(9*13) +....+…
2023
The sum of series represented as 1/(1*5) + 1/(5*9) + 1/(9*13) +....+ 1/(221*225) is
- A.
28/221
- B.
56/225
- C.
28/225
- D.
56/221
Attempted by 2 students.
Show answer & explanation
Correct answer: B
Answer: 56/225
Key idea: express each term using partial fractions so the series telescopes.
Write the general term. The denominators follow 1, 5, 9, ..., 221, 225, i.e. numbers of the form 4k+1 and 4k+5. So the kth term is 1/[(4k+1)(4k+5)] with k = 0,1,...,55.
Use partial fractions: 1/[(4k+1)(4k+5)] = (1/4)(1/(4k+1) - 1/(4k+5)).
Sum from k = 0 to 55. The terms telescope: most intermediate terms cancel, leaving (1/4)(1/1 - 1/225).
Simplify: (1/4)(1 - 1/225) = (1/4)(224/225) = 224/900 = 56/225.
Therefore the sum of the series is 56/225.