Class 11 Maths Chapter 8 Sequences and Series – Exercise 8.2 NCERT Solutions

Introduction

Exercise 8.2 focuses on the sum of terms in an arithmetic progression (AP). You will learn how to derive the formula for the sum of first $n$ terms, apply it to practical problems, and solve questions involving partial sums. This exercise builds the foundation for advanced series and summation techniques.

Key Concepts

1. Sum of First $n$ Terms of AP:

$$S_n=\frac{n}{2}[2a+(n-1)d]$$

or equivalently,

$$S_n=\frac{n}{2}(a+l)$$

where $a$ = first term, $d$ = common difference, $l$ = last term.

2. Special Cases:

   • If $d=0$, sum = $n\cdot a$.

   • If $d>0$, AP is increasing; if $d<0$, AP is decreasing.

NCERT Questions with Solutions (10)

Q1. Find sum of first 10 terms of AP: 2, 7, 12, …

$$a=2,d=5,n=10$$

$$S_{10}=\frac{10}{2}[2\cdot 2+(10-1)\cdot 5]=5[4+45]=245$$

Q2. Find sum of first 20 terms of AP: 1, 4, 7, …

$$a=1,d=3,n=20$$

$$S_{20}=\frac{20}{2}[2\cdot 1+(20-1)\cdot 3]=10[2+57]=590$$

Q3. Find sum of first 15 terms of AP: 3, 8, 13, …

$$a=3,d=5,n=15$$

$$S_{15}=\frac{15}{2}[2\cdot 3+(15-1)\cdot 5]=\frac{15}{2}[6+70]=\frac{15}{2}\cdot 76=570$$

Q4. Find sum of first 25 terms of AP: 10, 20, 30, …

$$a=10,d=10,n=25$$

$$S_{25}=\frac{25}{2}[20+240]=\frac{25}{2}\cdot 260=3250$$

Q5. Find sum of first 12 terms of AP: 7, 13, 19, …

$$a=7,d=6,n=12$$

$$S_{12}=\frac{12}{2}[14+66]=6\cdot 80=480$$

Q6. Find sum of first 30 terms of AP: 5, 9, 13, …

$$a=5,d=4,n=30$$

$$S_{30}=\frac{30}{2}[10+116]=15\cdot 126=1890$$

Q7. Find sum of first 18 terms of AP: 4, 9, 14, …

$$a=4,d=5,n=18$$

$$S_{18}=\frac{18}{2}[8+85]=9\cdot 93=837$$

Q8. Find sum of first 50 terms of AP: 1, 6, 11, …

$$a=1,d=5,n=50$$

$$S_{50}=\frac{50}{2}[2+245]=25\cdot 247=6175$$

Q9. Find sum of first 22 terms of AP: 2, 5, 8, …

$$a=2,d=3,n=22$$

$$S_{22}=\frac{22}{2}[4+63]=11\cdot 67=737$$

Q10. Find sum of first 40 terms of AP: 3, 6, 9, …

$$a=3,d=3,n=40$$

$$S_{40}=\frac{40}{2}[6+117]=20\cdot 123=2460$$

FAQs (10)

1. What is formula for sum of AP? $S_n=\frac{n}{2}[2a+(n-1)d]$.

2. What is alternative formula for sum? $S_n=\frac{n}{2}(a+l)$.

3. What is last term of AP? $l=a+(n-1)d$.

4. What is sum if $d=0$? $S_n=n\cdot a$.

5. Is AP always increasing? Only if $d>0$.

6. Can AP be decreasing? Yes, if $d<0$.

7. What is sum of first $n$ natural numbers? $\frac{n(n+1)}{2}$.

8. What is sum of first $n$ odd numbers? $n^2$.

9. What is sum of first $n$ even numbers? $n(n+1)$.

10. Why is this exercise important? It builds skills in summation and series analysis.

Conclusion

Exercise 8.2 has 10 solved questions and 10 FAQs that strengthen your understanding of the sum of arithmetic progressions. This builds the foundation for advanced series and summation techniques in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 8 Sequences and Series – Exercise 8.2 NCERT Solutions