Series: arithmetic and geometric progressions, sums, binomial expansion

Series (Syllabus 1.6) – Arithmetic & Geometric Progressions, Binomial Expansion

1. Arithmetic Progression (AP)

1.1 Definition & $n^{\text{th}}$ term

• Sequence in which the difference between consecutive terms is constant – the common difference $d$.
• First term: $a$.
• General term: $t_n = a + (n-1)d$.

1.2 Sum of the first $n$ terms

Using $\Sigma$‑notation:

$$S_n = \sum_{k=1}^{n} t_k = \sum_{k=1}^{n}\bigl[a+(k-1)d\bigr]$$

Evaluating the arithmetic series (pairing the first and last terms) gives two equivalent closed‑form expressions:

$$S_n = \frac{n}{2}\bigl(2a+(n-1)d\bigr)\;=\;\frac{n}{2}\bigl(t_1+t_n\bigr)$$

1.3 Special cases

• $d=0$ → All terms equal $a$; $S_n = na$.
• $a=0$ → $ t_n = (n-1)d$ (the progression starts from zero).
• $d<0$ → The AP decreases; the formulae remain valid.

1.4 Example

Find the sum of the first 15 terms of the AP $4,\;9,\;14,\dots$.

1. Identify $a=4$, $d=5$.
2. Find the 15‑th term: $t_{15}=a+(15-1)d=4+14\cdot5=74$.
3. Apply the sum formula:

$$S_{15}= \frac{15}{2}\bigl(4+74\bigr)=\frac{15}{2}\times78=585.$$

2. Geometric Progression (GP)

2.1 Definition & $n^{\text{th}}$ term

• Sequence in which each term after the first is obtained by multiplying the preceding term by a constant – the common ratio $r$.
• First term: $a$.
• General term: $g_n = a\,r^{\,n-1}$.

2.2 Sum of the first $n$ terms (finite GP)

In $\Sigma$‑notation:

$$G_n = \sum_{k=1}^{n} a\,r^{\,k-1}=a\sum_{k=0}^{n-1} r^{k}$$

Evaluating the geometric series (for $req1$) gives:

$$G_n = a\,\frac{1-r^{\,n}}{1-r}\qquad(req1)$$

2.3 Special cases

• $r=1$ → All terms equal $a$; $G_n = na$.
• $a=0$ → Every term is $0$; $G_n = 0$.
• $r=-1$ → Terms alternate $a,-a,a,\dots$; use the same formula (note the sign of $r^{\,n}$).

2.4 Infinite GP – convergence

An infinite geometric series

$$\displaystyle\sum_{k=1}^{\infty} a\,r^{\,k-1}$$

converges **iff** $|r|<1$. When it converges, the sum to infinity is

$$G_{\infty}= \frac{a}{1-r}\qquad(|r|<1).$$

If $|r|\ge 1$ the series diverges (no finite sum).

2.5 Example (finite GP)

Find the sum of the first 6 terms of the GP $3,6,12,\dots$.

1. $a=3$, $r=2$.
2. Apply the finite‑sum formula:

$$G_6 = 3\,\frac{1-2^{6}}{1-2}=3\,\frac{1-64}{-1}=3\times63=189.$$

2.6 Example (infinite GP)

Evaluate $\displaystyle\sum_{n=1}^{\infty}\frac{2}{3^{\,n}}$.

• Rewrite as $a r^{\,n-1}$ with $a=\frac{2}{3}$ and $r=\frac13$.
• Since $|r|<1$, the series converges:

$$G_{\infty}= \frac{a}{1-r}= \frac{\frac{2}{3}}{1-\frac13}= \frac{\frac{2}{3}}{\frac23}=1.$$

3. Binomial Expansion

3.1 Binomial theorem (positive integer exponent)

For any non‑negative integer $n$:

$$ (x+y)^n = \sum_{k=0}^{n} \binom{n}{k}\,x^{\,n-k}y^{\,k} $$

where the binomial coefficient

$$\displaystyle\binom{n}{k}= \frac{n!}{k!\,(n-k)!},\qquad 0\le k\le n.$$

3.2 General term

The $(k+1)^{\text{th}}$ term (counting from $k=0$) is

$$T_{k+1}= \binom{n}{k}\,x^{\,n-k}y^{\,k}.$$

3.3 Pascal’s triangle

Coefficients $\binom{n}{k}$ are read directly from Pascal’s triangle. The first six rows are shown below.

n	Coefficients $\displaystyle\binom{n}{k}$
0	1
1	1 1
2	1 2 1
3	1 3 3 1
4	1 4 6 4 1
5	1 5 10 10 5 1

3.4 Example – expansion up to a specific term

Expand $(2x-3)^4$ up to the term in $x^2$.

1. Identify $x\to2x$, $y\to-3$, $n=4$.
2. Write the first three terms ($k=0,1,2$):

\[ \begin{aligned} (2x-3)^4 &= \binom{4}{0}(2x)^4(-3)^0 + \binom{4}{1}(2x)^3(-3)^1 + \binom{4}{2}(2x)^2(-3)^2 + \dots\\[4pt] &= 1\cdot16x^4 -4\cdot8x^3\cdot3 +6\cdot4x^2\cdot9 + \dots\\[4pt] &= 16x^4 -96x^3 +216x^2 +\dots \end{aligned} \]

4. Practice Questions

1. Find the $20^{\text{th}}$ term of the AP $7,\,12,\,17,\dots$.
2. Calculate the sum of the first 10 terms of the GP $5,\,15,\,45,\dots$.
3. Determine whether the infinite series $\displaystyle\sum_{n=1}^{\infty}\frac{2}{3^{\,n}}$ converges, and if so, find its sum.
4. Expand $(x+2)^5$ and state the coefficient of $x^3$.
5. Given that the sum of the first $n$ terms of an AP is $S_n = 3n^2 + 5n$, find the first term $a$ and the common difference $d$.

5. Summary of Key Formulae

Series type	General term	Sum of $n$ terms	Special notes
Arithmetic progression (AP)	$t_n = a + (n-1)d$	$S_n = \displaystyle\frac{n}{2}\bigl(2a+(n-1)d\bigr)=\displaystyle\frac{n}{2}(t_1+t_n)$	Linear growth; $d$ may be positive, negative or zero.
Geometric progression (GP)	$g_n = a\,r^{\,n-1}$	$G_n = a\,\displaystyle\frac{1-r^{\,n}}{1-r}\;(req1)$ $G_{\infty}= \displaystyle\frac{a}{1-r}\quad(|r|<1)$	Infinite sum exists only for $|r|<1$. $r=1$ gives $G_n=na$.
Binomial expansion	$T_{k+1}= \displaystyle\binom{n}{k}x^{\,n-k}y^{\,k}$	$(x+y)^n = \displaystyle\sum_{k=0}^{n}\binom{n}{k}x^{\,n-k}y^{\,k}$	Coefficients read from Pascal’s triangle; $n$ must be a non‑negative integer.