Exercise 4.3 — Question 1

Arithmetic Sequences and Series

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a fixed constant called the common difference $d$.

General term (nth term): $a_n=a_1+(n-1)d$

Sum of first $n$ terms: $S_n=\frac{n}{2}(2a_1+(n-1)d)=\frac{n}{2}(a_1+a_n)$

Geometric Sequences and Series

A geometric sequence is a sequence in which each term after the first is obtained by multiplying by a fixed constant called the common ratio $r$.

General term (nth term): $a_n=a_1\cdot r^{n-1}$

Sum of first $n$ terms ($r\ne 1$): $S_n=\frac{a_1(r^n-1)}{r-1}=\frac{a_1(1-r^n)}{1-r}$

Harmonic Sequences

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence.

If $a_1,a_2,a_3,\dots$ is a harmonic sequence, then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots$ is an arithmetic sequence.

nth term of a harmonic sequence: $a_n=\frac{1}{a_1^{-1}+(n-1)d}$ where $d$ is the common difference of the corresponding arithmetic sequence.

Identifying Sequences

Worked Examples

Example 1 — Arithmetic Series

Find the sum of the first 10 terms of the arithmetic sequence $3,7,11,15,\dots$

Solution:

• $a_1=3$, $d=4$, $n=10$ $S_{10}=\frac{10}{2}(2(3)+(10-1)(4))=5(6+36)=5\times 42=210$

Example 2 — Geometric Series

Find the sum of the first 5 terms of the geometric sequence $2,6,18,54,\dots$

Solution:

• $a_1=2$, $r=3$, $n=5$ $S_5=\frac{2(3^5-1)}{3-1}=\frac{2(243-1)}{2}=242$

Example 3 — Harmonic Sequence

Find the 5th term of the harmonic sequence $1,\frac{1}{3},\frac{1}{5},\dots$

Solution: The reciprocals $1,3,5,\dots$ form an AP with $a_1=1$, $d=2$. $a_5^{-1}=1+(5-1)(2)=9⟹a_5=\frac{1}{9}$