SATHEE: Sequences and Series 5 Question 19

Sequences and Series 5 Question 19

10. Let $S_{1}, S_{2}, \dots$ be squares such that for each $n \geq 1$ the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n + 1}$ . If the length of a side of $S_{1}$ is $10 c m$ , then for which of the following values of $n$ is the area of $S_{n}$ less than $1 s q c m$ ?

(1999, 3M)

(a) 7

(b) 8

(d) 10

Analytical & Descriptive Questions

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Solution:

Let $S_{10}$ be the sum of first ten terms of the series. Then, we have

$S_{10} = 1 {\frac{3}{5}}^{2} + 2 {\frac{2}{5}}^{2} + 3 {\frac{1}{5}}^{2} + 4^{2} + 4 {\frac{4}{5}}^{2}$

$+ \dots$ to 10 terms

$\begin{aligned} = {\frac{8}{5}}^{2} + {\frac{12}{5}}^{2} + {\frac{16}{5}}^{2} + 4^{2} + {\frac{24}{5}}^{2} + \dots to 10 terms \\ = \frac{1}{5^{2}} (8^{2} + 12^{2} + 16^{2} + 20^{2} + 24^{2} + \dots to 10 terms) \\ = \frac{4^{2}}{5^{2}} (2^{2} + 3^{2} + 4^{2} + 5^{2} + \dots to 10 terms) \\ = \frac{4^{2}}{5^{2}} (2^{2} + 3^{2} + 4^{2} + 5^{2} + \dots + 11^{2}) \\ = \frac{16}{25} ((1^{2} + 2^{2} + \dots + 11^{2}) - 1^{2}) \\ = \frac{16}{25} \frac{11 \cdot (11 + 1) (2 \cdot 11 + 1)}{6} - 1 \\ = \frac{16}{25} (506 - 1) = \frac{16}{25} \times 505 \Rightarrow \frac{16}{5} m = \frac{16}{25} \times 505 = 101 \end{aligned}$

Sequences and Series 5 Question 19

10. Let $S_{1}, S_{2}, \dots$ be squares such that for each $n \geq 1$ the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n + 1}$ . If the length of a side of $S_{1}$ is $10 c m$ , then for which of the following values of $n$ is the area of $S_{n}$ less than $1 s q c m$ ?

Analytical & Descriptive Questions

Solution:

Engineering Preparation

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NCERT Books & Solutions

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Other Resources

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Sample Mock Test

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NCERT Exercise Video Solution

Sequences and Series 5 Question 19

10. Let S1,S2,… be squares such that for each n≥1 the length of a side of Sn equals the length of a diagonal of Sn+1. If the length of a side of S1 is 10cm, then for which of the following values of n is the area of Sn less than 1sqcm ?

Analytical & Descriptive Questions

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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10. Let $S_{1}, S_{2}, \dots$ be squares such that for each $n \geq 1$ the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n + 1}$ . If the length of a side of $S_{1}$ is $10 c m$ , then for which of the following values of $n$ is the area of $S_{n}$ less than $1 s q c m$ ?