Class 11 Maths Chapter 8 Sequences and Series – Exercise 8.3 NCERT Solutions

Introduction

Exercise 8.3 introduces the concept of geometric progression (GP). You will learn how to identify GPs, find common ratios, calculate general terms, and evaluate sums of finite and infinite geometric series. This exercise builds the foundation for advanced series and summation techniques in mathematics.

Key Concepts

1. Geometric Progression (GP): A sequence where each term after the first is obtained by multiplying with a constant ratio.

$$a,ar,a{r}^2,a{r}^3,\dots$$

where $a$ = first term, $r$ = common ratio.

2. General Term of GP:

$$a_n=a\cdot r^{n-1}$$

3. Sum of First $n$ Terms of GP:

$$S_n=\frac{a(r^n-1)}{r-1},r\mathrm{\ne }1$$

4. Sum of Infinite GP: If $\mathrm{\mid }r\mathrm{\mid }<1$:

$$S_{\mathrm{\infty }}=\frac{a}{1-r}$$

NCERT Questions with Solutions (10)

Q1. Find 7th term of GP: 2, 6, 18, …

$$a=2,r=3,a_7=2\cdot 3^6=1458$$

Q2. Find 10th term of GP: 5, 10, 20, …

$$a=5,r=2,a_{10}=5\cdot 2^9=2560$$

Q3. Find 12th term of GP: 3, 9, 27, …

$$a=3,r=3,a_{12}=3\cdot 3^{11}=3\cdot 177147=531441$$

Q4. Find sum of first 8 terms of GP: 2, 4, 8, …

$$a=2,r=2,S_8=\frac{2(2^8-1)}{2-1}=2(255)=510$$

Q5. Find sum of first 6 terms of GP: 3, 6, 12, …

$$a=3,r=2,S_6=\frac{3(2^6-1)}{2-1}=3(63)=189$$

Q6. Find sum of first 5 terms of GP: 1, 2, 4, …

$$a=1,r=2,S_5=\frac{1(2^5-1)}{2-1}=31$$

Q7. Find sum of first 10 terms of GP: 5, 15, 45, …

$$a=5,r=3,S_{10}=\frac{5(3^{10}-1)}{3-1}=\frac{5(59049-1)}{2}=147620$$

Q8. Find sum of infinite GP: 1, \tfrac{1}{2}, \tfrac{1}{4}, …

$$a=1,r=\frac{1}{2},S_{\mathrm{\infty }}=\frac{1}{1-\frac{1}{2}}=2$$

Q9. Find sum of infinite GP: 3, 1, \tfrac{1}{3}, …

$$a=3,r=\frac{1}{3},S_{\mathrm{\infty }}=\frac{3}{1-\frac{1}{3}}=\frac{3}{\frac{2}{3}}=\frac{9}{2}$$

Q10. Find sum of infinite GP: 10, 5, 2.5, …

$$a=10,r=\frac{1}{2},S_{\mathrm{\infty }}=\frac{10}{1-\frac{1}{2}}=20$$

FAQs (10)

1. What is geometric progression? Sequence with constant ratio between terms.

2. What is general term of GP? $a_n=a\cdot r^{n-1}$.

3. What is formula for sum of GP? $S_n=\frac{a(r^n-1)}{r-1}$.

4. What is sum of infinite GP? $S_{\mathrm{\infty }}=\frac{a}{1-r}$ if $\mathrm{\mid }r\mathrm{\mid }<1$.

5. What is common ratio? Ratio between consecutive terms.

6. Can GP have negative ratio? Yes, terms alternate in sign.

7. Can GP have ratio 1? Yes, then all terms equal.

8. Is GP always increasing? Not necessarily; depends on $r$.

9. What is relation between AP and GP? AP uses addition, GP uses multiplication.

10. Why is this exercise important? It builds skills in handling geometric progressions and infinite series.

Conclusion

Exercise 8.3 has 10 solved questions and 10 FAQs that strengthen your understanding of geometric progressions and their sums. This completes the Sequences and Series chapter in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 8 Sequences and Series – Exercise 8.3 NCERT Solutions