4.8 Arithmetic Progression

Arithmetic Sequence

A sequence $a_1,a_2,a_3,\dots$ is called an arithmetic sequence (or arithmetic progression, AP) if the difference between any two consecutive terms is constant.

This constant difference is called the common difference $d$: $d=a_{n+1}-a_n\text{for all }n\ge 1$

General Term (nth Term)

The general term of an arithmetic sequence is: $a_n=a_1+(n-1)d$

where:

• $a_1$ = first term
• $d$ = common difference
• $n$ = term number

Example: Find the 10th term of the AP: $3,7,11,15,\dots$

Here $a_1=3$, $d=4$. $a_{10}=3+(10-1)(4)=3+36=39$

Sum to n Terms of an Arithmetic Series

The sum of the first $n$ terms of an arithmetic series is: $S_n=\frac{n}{2}[2a_1+(n-1)d]$

Alternatively, if the last term $a_n=l$ is known: $S_n=\frac{n}{2}(a_1+l)$

Example: Find the sum of the first 20 terms of $2+5+8+11+\cdots$

Here $a_1=2$, $d=3$, $n=20$. $S_{20}=\frac{20}{2}[2(2)+(20-1)(3)]=10[4+57]=10\times 61=610$

Sigma Notation

Sigma notation $\sum$ is used to write series compactly.

${\sum }_{k=1}^n{a}_k=a_1+a_2+a_3+\cdots +a_n$

For an arithmetic series with first term $a_1$ and common difference $d$: ${\sum }_{k=1}^n[a_1+(k-1)d]=\frac{n}{2}[2a_1+(n-1)d]$

Example: Evaluate $\sum _{k=1}^5(3k-1)$

$=2+5+8+11+14=40$

Using the formula: $a_1=2$, $d=3$, $n=5$: $S_5=\frac{5}{2}[2(2)+4(3)]=\frac{5}{2}[4+12]=\frac{5}{2}(16)=40✓$

Standard Summation Formulas

These are essential formulas for evaluating series using sigma notation.

Sum of First n Natural Numbers

${\sum }_{k=1}^{n}k=1+2+3+\cdots +n=\frac{n(n+1)}{2}$

Example: $\sum _{k=1}^{10}k=\frac{10\times 11}{2}=55$

Sum of Squares of First n Natural Numbers

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

Example: $\sum _{k=1}^5{k}^2=\frac{5\times 6\times 11}{6}=55$

Sum of Cubes of First n Natural Numbers

${\sum }_{k=1}^n{k}^3=1^3+2^3+3^3+\cdots +n^3={\left[\frac{n(n+1)}{2}\right]}^2$

Example: $\sum _{k=1}^4{k}^3={\left[\frac{4\times 5}{2}\right]}^2=10^2=100$

Useful Properties of Sigma Notation

${\sum }_{k=1}^{n}c=nc(c\text{ is a constant})$

${\sum }_{k=1}^{n}c{a}_k=c{\sum }_{k=1}^n{a}_k$

${\sum }_{k=1}^n(a_k\pm b_k)={\sum }_{k=1}^n{a}_k\pm {\sum }_{k=1}^n{b}_k$

Example: Evaluate $\sum _{k=1}^n(2k^2+3k+1)$

$=2{\sum }_{k=1}^n{k}^2+3{\sum }_{k=1}^{n}k+{\sum }_{k=1}^{n}1$

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

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