1.

1). 2225/137282). 4883/171603). 1893/102484). 4933/17160

Answer»

The given SERIES can be written as,

$(\RIGHTARROW \frac{1}{{3 \times 5}} + \frac{1}{{1 \times 4 \times 7}} + \frac{1}{{5 \times 7}} + \frac{1}{{4 \times 7 \times 10}} + \frac{1}{{7 \times 9}} + \frac{1}{{7 \times 10 \times 13}} + \frac{1}{{9 \times 11}} + \frac{1}{{10 \times 13 \times 16}})$

⇒ 1/2 × [1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11] + 1/6[(7 - 1)/(1 × 4 × 7) + (10 - 4)/(4 × 7 × 10) + (13 - 7)/(7 × 10 × 13) + (16 - 10)/(10 × 13 × 16)]

⇒ 1/2[1/3 - 1/11] + 1/6[1/4 - 1/208]

⇒ 4/33 + 17/416 = 2225/13728

$(\frac{1}{{{4^2} - 1}} + \frac{1}{{1 \times 4 \times 7}} + \frac{1}{{{6^2} - 1}} + \frac{1}{{4 \times 7 \times 10}} + \frac{1}{{{8^2} - 1}} + \frac{1}{{7 \times 10 \times 13}} + \frac{1}{{{{10}^2} - 1}} + \frac{1}{{10 \times 13 \times 16}})$


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