Question

The sum to infinity of $\dfrac17+\dfrac{2}{7^2}+\dfrac{1}{7^3}+\dfrac{2}{7^4}+\cdots$ is:

$\dfrac15$

$\dfrac1{24}$

$\dfrac5{48}$

$\dfrac1{16}$

None of these

Answer: E

Concepts:geometric sequence summation

Difficulty rating: 1670

Small Hint:

Group the series into consecutive pairs of terms

Big Hint:

Each pair is $\frac{1}{49}$ times the preceding pair

Solution:

Grouping consecutive terms gives a geometric series whose first grouped term is $\frac17+\frac2{49}=\frac9{49}$ and whose ratio is $\tfrac1{49}.$ Hence the sum is $\frac{\frac9{49}}{1-\frac1{49}} =\frac9{48} =\frac3{16}.$ This is not among choices A through D.

Thus, the correct answer is E.

1950 AMC 12 Problem 43

Answer: E

Difficulty rating: 1670

Small Hint:

Big Hint:

Solution:

Problem 43 in Other Years