If S is sum of (5/12 ⋅ 22 ⋅ 3)+(8/22 ⋅ 32 ⋅ 4)+(11/32 ⋅ 42 ⋅ 5)+(14/42 ⋅ 52 ⋅ 6)+ ldots ldots ∞ terms, then S is equal to

Question

If S is sum of (5/12 ⋅ 22 ⋅ 3)+(8/22 ⋅ 32 ⋅ 4)+(11/32 ⋅ 42 ⋅ 5)+(14/42 ⋅ 52 ⋅ 6)+ ldots ldots ∞ terms, then S is equal to (A) 1 (B) (1/2) (C) (1/4) (D) (3/4)

Tardigrade Admin · Accepted Answer

Θ Tn=((3 n+2)/n2(n+1)2(n+2))=((n2+3 n+2)-n2/n2(n+1)2(n+2)) =(1/ n 2( n +1))-(1/( n +1)2( n +2)) ∴ S =((1/12 ⋅ 2)-(1/22 ⋅ 3))+((1/22 ⋅ 3)-(1/32 ⋅ 4))+((1/32 ⋅ 4)-(1/42 ⋅ 5)) ldots ldots ∞ text terms =(1/2)

Q. If $S$ is sum of $\frac{5}{1 ^{2} \cdot 2 ^{2} \cdot 3} + \frac{8}{2 ^{2} \cdot 3 ^{2} \cdot 4} + \frac{11}{3 ^{2} \cdot 4 ^{2} \cdot 5} + \frac{14}{4 ^{2} \cdot 5 ^{2} \cdot 6} + \dots\dots \infty$ terms, then $S$ is equal to

1

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{3}{4}$

Q. If S is sum of 12⋅22⋅35​+22⋅32⋅48​+32⋅42⋅511​+42⋅52⋅614​+……∞ terms, then S is equal to

1

21​

41​

43​

ΘTn​=n2(n+1)2(n+2)(3n+2)​=n2(n+1)2(n+2)(n2+3n+2)−n2​ =n2(n+1)1​−(n+1)2(n+2)1​ ∴S=(12⋅21​−22⋅31​)+(22⋅31​−32⋅41​)+(32⋅41​−42⋅51​)……∞ terms =21​

Q. If $S$ is sum of $\frac{5}{1 ^{2} \cdot 2 ^{2} \cdot 3} + \frac{8}{2 ^{2} \cdot 3 ^{2} \cdot 4} + \frac{11}{3 ^{2} \cdot 4 ^{2} \cdot 5} + \frac{14}{4 ^{2} \cdot 5 ^{2} \cdot 6} + \dots\dots \infty$ terms, then $S$ is equal to

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{3}{4}$