A sequence a_1, a_2, a_3, \dots is defined by a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}} for al… — GRE General Test

A sequence $a_1, a_2, a_3, \dots$ is defined by $a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}}$ for all positive integers $n$ . If $S = \sum_{n=1}^{98} a_n$ , which of the following is closest to the value of $S$ when rounded to the nearest tenth?

8.8Answer
B
4.5
C
9.0
D
9.9
E
17.5

Answer

The value of

S

rounded to the nearest tenth is 8.8.

The term

a_n

simplifies to

\frac{\sqrt{n+2} - \sqrt{n}}{2}

upon rationalizing the denominator. Summing from

n=1

98

causes all middle terms to cancel out, leaving

\frac{\sqrt{99} + \sqrt{100} - \sqrt{1} - \sqrt{2}}{2}

. Substituting

\sqrt{100} = 10

\sqrt{1} = 1

\sqrt{99} \approx 9.95

, and

\sqrt{2} \approx 1.41

gives approximately

8.77

, which rounds to 8.8.

Step-by-Step Solution

Rationalize the general term

a_n

a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}} \cdot \frac{\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} - \sqrt{n}} = \frac{\sqrt{n+2} - \sqrt{n}}{(n+2) - n} = \frac{\sqrt{n+2} - \sqrt{n}}{2}

Eliminating radicals from the denominator reveals the underlying telescoping structure of the sequence.

Expand the summation

S = \sum_{n=1}^{98} a_n

S = \frac{1}{2} \left[ (\sqrt{3} - \sqrt{1}) + (\sqrt{4} - \sqrt{2}) + (\sqrt{5} - \sqrt{3}) + \dots + (\sqrt{99} - \sqrt{97}) + (\sqrt{100} - \sqrt{98}) \right]

Writing out initial and final terms demonstrates which terms cancel.

Simplify the telescoping sum

S = \frac{\sqrt{99} + \sqrt{100} - \sqrt{1} - \sqrt{2}}{2} = \frac{\sqrt{99} + 10 - 1 - \sqrt{2}}{2} = \frac{9 + \sqrt{99} - \sqrt{2}}{2}

All intermediate terms cancel out, leaving two positive boundary terms and two negative boundary terms.

Estimate square root values and perform rounding

Since

\sqrt{99} \approx 9.94987

and

\sqrt{2} \approx 1.41421

S \approx \frac{9 + 9.94987 - 1.41421}{2} = \frac{17.53566}{2} = 8.76783 \approx 8.8

Evaluating the radicals to two decimal places allows accurate rounding to the nearest tenth.

Key Concept

Telescoping Series Summation and Square Root Estimation

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