# Chapter 06: Sequences and Series

Notes (Solutions) of Chapter 06: Sequences and Series, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore.

### Contents & summary

- Introduction
- Types of Sequences
- Exercise 6.1

- Arithmetic Progression(A.P)
- Exercise 6.2

- Aritmetic Mean (A.M)
*n*Arithmetic Means between two given Numbers- Exercise 6.3

- Series
- Exercise 6.4

- World Problems on A.P.
- Exercise 6.5

- Geometric Progression(G.P)
- Exercise 6.6

- Geometric Means
*n*Geometric Means between two given numbers- Exercise 6.7

- Sum of
*n*terms of a Geometric Series - The Infinite Geometric Series
- Exercise 6.8

- World Problems on G.P.
- Exercise 6.9

- Harmonic Progression(H.P)
- Harmonic Means
*n*Harmonic Means between two numbers

- Relation between Arithmic, Geometric and Harmonic Means
- Exercise 6.10

- Sigma Notation(or Summation Notation)
- To find the Formulae of Sum
- Exercise 6.11

### Some corrected solutions

**Exercise 6.2⇒ Question 11(i)**

If $l,m,n$ are the $p$th, $q$th, $r$th terms of an A.P., show that
$$l(q-r)+m(r-p)+n(p-q)=0$$
**Solution:** Let $a_1$ be first term and $d$ be common difference of A.P, then
$$\begin{align}l=a_1+(p-1)d,\\ m=a_1+(q-1)d,\\ n=a_1+(r-1)d.\end{align}$$
Now $$\begin{align}L.H.S &= l(q-r)+m(r-p)+n(p-q)\\
&= lq-lr+mr-mp+np-nq\\
&=(l-n)q+(m-l)r+(n-m)p\\
&=(a_1+(p-1)d-a_1-(r-1)d)q+( a_1+(q-1)d-a_1-(p-1)d)r\\
&\,\,+(a_1+(r-1)d-a_1-(q-1)d)p\\
&=((p-1)d-(r-1)d)q+( (q-1)d-(p-1)d)r+((r-1)d-(q-1)d)p\\
&=(p-1-r+1)dq+(q-1-p+1)dr+(r-1-q+1)dp\\
&=(p-r)dq+(q-p)dr+(r-q)dp\\
&=[pq-qr+qr-pr+pr-pq]d=(0)d=0=R.H.S\end{align}$$

**Exercise 6.4⇒ Question 3(ii)**

An equation $3n^2-17n-288=0$ should be $3n^2-17n-228=0$. … *suggested by Abdullah Zafar (Garrison Academy Kharian)*

### Solutions

- Exercise 6.1 | View Online | Download PDF (115KB)

- Exercise 6.2 | View Online | Download PDF (409KB)

- Exercise 6.3 | View Online | Download PDF (123KB)

- Exercise 6.4 | View Online | Download PDF (757KB)

- Exercise 6.5 | View Online | Download PDF (124KB)

- Exercise 6.6 | View Online | Download PDF (332KB)

- Exercise 6.7 | View Online | Download PDF (220KB)

- Exercise 6.8 | View Online | Download PDF (389KB)

- Exercise 6.9 | View Online | Download PDF (115KB)

- Exercise 6.10 by Momin Ali |View Online | Download PDF (560KB)

- Exercise 6.10 |View Online | Download PDF (365KB)

- Exercise 6.11 | View Online | Download PDF (162KB)

### Question Bank by Prof. Nusni Natasha

- Chapter 06 (Question Bank) | View Online | Download PDF (410KB)

### Short Questions

The following short questions was send by Mr. Akhtar Abbas.

- Chapter 06 | View Online | Download PDF