Question

Evaluate \[ \left(\frac{4}{7}+\frac{1}{3}\right)+\left(\frac{4}{7}+\frac{4}{3}\right)\left(\frac{1}{3}\right) +\left(\frac{4}{7}+\frac{4}{3}\right)^2\left(\frac{1}{3}\right)^2+\cdots \]

Hint:

Always rewrite long series in standard G.P. form before applying formulas.

Answer 1

Correct Answer: 2.5

Step 1: Recognize the type of series

The given expression represents an infinite geometric progression with common ratio

r = 1/3

and first term

a = 4/7 + 1/3

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Step 2: Compute the first term

a = (12 + 7) / 21

a = 19 / 21

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Step 3: Use the sum formula for an infinite G.P.

Sum of an infinite geometric series is given by:

S = a / (1 − r)

Substituting the values,

S = (19 / 21) ÷ (2 / 3)

S = 19 / 14

Which gives,

S = 5 / 2

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Final Answer:

The value of the given expression is
5 / 2