Class 11 Maths Chapter 8 Sequences and Series – Exercise 8.4 NCERT Solutions

Introduction

Exercise 8.4 focuses on the sum to n terms of special series. Students learn how to evaluate sums of series like $\sum n$, $\sum n^2$, and $\sum n^3$, and apply these formulas to solve problems. This exercise is essential for building problem‑solving skills in sequences and series.

Key Formulas

1. Sum of first $n$ natural numbers:

$$1+2+3+\cdots +n=\frac{n(n+1)}{2}$$

2. Sum of squares of first $n$ natural numbers:

$$1^2+2^2+3^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$$

3. Sum of cubes of first $n$ natural numbers:

$$1^3+2^3+3^3+\cdots +n^3={\left(\frac{n(n+1)}{2}\right)}^2$$

Common Mistakes

• Forgetting to apply the correct formula for squares or cubes.

• Mixing up $\sum n^2$ and $\sum n^3$.

• Not simplifying expressions properly.

• Ignoring factorization in final answers.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find sum of first 10 natural numbers.

$$S=\frac{10(10+1)}{2}=55$$

Q2. Find sum of squares of first 7 natural numbers.

$$S=\frac{7(7+1)(2\cdot 7+1)}{6}=\frac{7\cdot 8\cdot 15}{6}=140$$

Q3. Find sum of cubes of first 5 natural numbers.

$$S={\left(\frac{5(5+1)}{2}\right)}^2={\left(\frac{30}{2}\right)}^2={15}^2=225$$

Q4. Evaluate ${\sum }_{k=1}^{20}k$.

$$S=\frac{20(21)}{2}=210$$

Q5. Evaluate ${\sum }_{k=1}^{15}{k}^2$.

$$S=\frac{15(16)(31)}{6}=1240$$

Q6. Evaluate ${\sum }_{k=1}^{12}{k}^3$.

$$S={\left(\frac{12(13)}{2}\right)}^2=(78{)}^2=6084$$

Q7. Find sum of first 50 natural numbers.

$$S=\frac{50(51)}{2}=1275$$

Q8. Find sum of squares of first 10 natural numbers.

$$S=\frac{10(11)(21)}{6}=385$$

Q9. Find sum of cubes of first 8 natural numbers.

$$S={\left(\frac{8(9)}{2}\right)}^2=(36{)}^2=1296$$

Q10. Evaluate ${\sum }_{k=1}^n{k}^3$ for $n=20$.

$$S={\left(\frac{20(21)}{2}\right)}^2=(210{)}^2=44100$$

FAQs (10)

FAQ1. What is formula for $\sum n$? $\frac{n(n+1)}{2}$.

FAQ2. What is formula for $\sum n^2$? $\frac{n(n+1)(2n+1)}{6}$.

FAQ3. What is formula for $\sum n^3$? ${\left(\frac{n(n+1)}{2}\right)}^2$.

FAQ4. Why are these formulas important? They simplify long summations.

FAQ5. Can these formulas be derived? Yes, using induction or algebraic identities.

FAQ6. What is difference between $\sum n^2$ and $\sum n^3$? Squares vs cubes of natural numbers.

FAQ7. Do these formulas work for all $n$? Yes, for positive integers.

FAQ8. What is practical use of these sums? Used in statistics, physics, and computer science.

FAQ9. What is shortcut for sum of cubes? Square of sum of first $n$ natural numbers.

FAQ10. Why is Exercise 8.4 important? It builds foundation for advanced series problems.

Conclusion

Exercise 8.4 covers special series sums with solved examples and FAQs. Mastering these formulas helps in solving complex problems in sequences and series.

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New Syllabus-Class 11 Maths Chapter 8 Sequences and Series – Exercise 8.4 NCERT Solutions