2006 iTest Problems/Problem 31
Problem
The value of the infinite series can be expressed as
where
and
are relatively prime positive numbers. Compute
.
Solution
Notice that , and notice that the numerator contains
. Also, notice that
. We can rewrite the expression by factoring and splitting the fractions.
In the fraction
, we can split the fraction as
, making
a telescoping series. Plugging in values results in
.
In the fraction , plugging in the first few values results in
. The structure suggests that we could try to model the same fraction with a telescoping series.
Substituting for
in
results in
. Also, because
, we find that the expression is equal to
, confirming that
is a telescoping series. The expression equals
.
Therefore, , so
.
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem 30 |
Followed by: Problem 32 | |
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