Find the sum of the following series (with infinite terms) 2 \\sqrt{2}, {4}/{\\sqrt{3}, {4 \\sqrt{2}/{3} ..............

2 \\sqrt{3}(\\sqrt{3}+\\sqrt{2})

2 \\sqrt{3}(\\sqrt{3}-\\sqrt{2})

2 \\sqrt{6}(\\sqrt{3}+\\sqrt{2})

2 \\sqrt{6}(\\sqrt{3}-\\sqrt{2)}

CMAT 2021 Solved Question Paper

Question : 24 of 100

Marks: +1, -0

Find the sum of the following series (with infinite terms)

2 \√{2}, {4}/{\√{3}}, {4 \√{2}}/{3} ..............

Solution:

Given series,

2 \√{2}, {4}/{\√{3}}, {4 \√{2}}/{3} ..............

∵ Common ratio

(r)={{4}/{\√{3}}}/{2 \√{2}}={{4 \√{2}}/{3}}/{{4}/{\√{3}}}={\√{2}}/{\√{3}}

∴ This is a geometric progression with first term(a)

=2 \√{2}

and common ratio

(\γ)={\√{2}}/{\√{3}}

Now, we know sum of an infinite GP,

\S_{\∞}={a}/{1-r}

={2 \√{2}}/{1-{\√{2}}/{\√{3}}}={2 \√{3} ×\√{2}}/{\√{3}-√{\2}}

={2 \√{6}}/{\√{3}-\√{2}}

={2 \√{6}}/{(\√{3}-\√{2})} ×{(\√{3}+\√{2})}/{(\√{3}+\√{2})}

={2 \√{6}(\√{3}+\√{2})}/{3-2}=2 \√{6} \×(\√{3}+\√{2})

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