Question

1/6+1/12+1/20+1/30•••••••••+1/9900=?

Answer 1

Answer:

The value of the given expression is: $\dfrac{49}{100}$.

Step-by-step explanation:

We need to find the value of the expression:

$=\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+.....+\dfrac{1}{9900}\\\\=\dfrac{1}{2\times3}+\dfrac{1}{3\times 4}+\dfrac{1}{4\times5}+\dfrac{1}{5\times6}+.....+\dfrac{1}{99\times100}\\\\=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1`}{4}+\dfrac{1}{4}-\dfrac{1}{5}+....+\dfrac{1}{99}-\dfrac{1}{100}\\\\=\dfrac{1}{2}-\dfrac{1}{100\\}\\\\=\dfrac{49}{100}$

Hence, the value of the given expression is: $\dfrac{49}{100}$.

Answer 2

99/100

Step-by-step explanation:

Given,

1/(1×2) + 1/(2×3) = 1/2 + 1/6

= (3 + 1)/6

= 4/6

= 2/3

Now,

1/(1×2) + 1/(2×3) + 1/(3×4) = 2/3 + 1/(3×4)

= 2/3 + 1/12

= (8 + 1)/12

= 9/12  or 3/4

Similarly,

1/(1×2) + 1/(2×3) + 1/(3×4) + 1/(4 *5) = 3/4 + 1/(4×5)

= 3/4 + 1/20

= (15 + 1)/20

= 16/20  or 4/5

We can notice that the pattern followed is:  

A = 1/(1×2) +1/(2×3) + 1/(3×4) + ... + 1/{(n)×(n+1)}  = n/(n + 1)  

So,

1/(1×2) + 1/(2×3) + 1/(3×4) + 1/(4 *5)...+1/(99×100)

⇒ 1/6+1/12+1/20+1/30••• + 1/9900 = 99/100    (∵ n/(n + 1)

Learn more: Solve the equation

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