Sequences and Series – Arithmetic & Geometric Progressions

1. Sequences and Series

A sequence is an ordered list of numbers $\{a_1,a_2,a_3,\dots \}$.
A series is the sum of the terms of a sequence:
$$S_n=a_1+a_2+\cdots +a_n.$$
Notation: $a_n$ denotes the $n^{th}$ term; $S_n$ the sum of the first $n$ terms.

Example: The sequence of even numbers $2,4,6,8,\dots$ has $a_n=2n$.

2. Arithmetic Progression (AP)

2.1 Definition

An AP is a sequence where the difference between consecutive terms is constant:
$$a_{n+1}-a_n=d(\text{common difference}).$$

2.2 nth Term Formula

$$a_n=a_1+(n-1)d.$$

2.3 Sum of First $n$ Terms

$$S_n=\frac{n}{2}(2a_1+(n-1)d).$$

2.4 Worked Example

Find $S_{10}$ for an AP with $a_1=3$ and $d=2$.

1. Compute $a_{10}=3+(10-1)\times 2=3+18=21$.
2. Use sum formula:
   $$S_{10}=\frac{10}{2}(2\times 3+(10-1)\times 2)=5(6+18)=5\times 24=120.$$

2.5 Visualisation – Graph of AP

2.6 Visualisation – Number Line for AP

3. Geometric Progression (GP)

3.1 Definition

A GP is a sequence where the ratio between consecutive terms is constant:
$$\frac{a_{n+1}}{a_n}=r(\text{common ratio}).$$

3.2 nth Term Formula

$$a_n=a_1{r}^{n-1}.$$

3.3 Sum of First $n$ Terms

$$S_n=a_1\frac{1-r^n}{1-r}(r\ne 1).$$

3.4 Sum to Infinity

If $|r|<1$, the infinite series converges:
$$S_{\infty }=\frac{a_1}{1-r}.$$

3.5 Worked Example

Find $S_5$ for a GP with $a_1=2$ and $r=3$.

1. Compute terms: $2,6,18,54,162$.
2. Sum:
   $$S_5=2\frac{1-3^5}{1-3}=2\frac{1-243}{-2}=2\frac{-242}{-2}=2\times 121=242.$$

3.6 Visualisation – Graph of GP

3.7 Visualisation – Bar Chart of Partial Sums

4. Comparison of AP and GP

4.1 Advantages & Disadvantages

• AP: Simple to compute; models linear processes; no convergence issues.
• GP: Captures exponential behaviour; useful for finance and technology; requires $|r|<1$ for convergence.

5. Real‑World Applications

5.1 Bank Loan Interest – Compound Interest (GP)

A loan of ₹1,000,000 at 7 % annual interest compounded yearly.
Principal $P=1,000,000$, rate $r=1.07$.
Amount after $n$ years: $A_n=P{r}^n$.
After 5 years:
$$A_5=1,000,000\times 1.07^5\approx 1,000,000\times 1.40255=1,402,550.$$
The interest earned is $A_5-P=402,550$.

5.2 Daraz Order Queue – Processing Time (AP)

Each order takes 2 minutes to process.
Time to process first 10 orders:
$$S_{10}=\frac{10}{2}(2\times 2+(10-1)\times 2)=5(4+18)=5\times 22=110\text{ minutes}.$$

5.3 eSewa Transaction Fees – Fixed + Variable (AP)

eSewa charges a fixed fee of ₹5 plus 1 % of the transaction amount.
For a ₹200 transaction:
$$\text{Fee}=5+0.01\times 200=5+2=7\text{ rupees}.$$
The variable part grows linearly with amount, an AP in the fee structure.

5.4 Google Search Ranking – Exponential Decay (GP)

Page relevance score decays exponentially with distance from the query term.
If initial score is 1 and decay factor $r=0.8$, after 3 steps:
$$S_3=1\times 0.8^3=\mathrm{0.512.}$$

6. Process Flowcharts

6.1 Sum of an AP – Flowchart

flowchart TD
    "Start" --> "Input a1, d, n"
    "Compute a_n = a1 + (n-1)d" --> "Compute S_n = n/2*(2*a1 + (n-1)*d)"
    "Output S_n" --> "End"

6.2 Sum of a GP – Flowchart

flowchart TD
    "Start" --> "Input a1, r, n"
    "Check r ≠ 1" --> "Compute S_n = a1*(1 - r^n)/(1 - r)"
    "If |r|<1 and n→∞" --> "Compute S∞ = a1/(1 - r)"
    "Output S_n or S∞" --> "End"

7. In the real world

• Bank loan amortization schedule – shows how each payment reduces principal and interest.

• eSewa transaction fee chart – illustrates fixed and variable components of fees.

• Daraz order queue diagram – visualises sequential processing of orders.

• Google search result ranking – algorithm uses exponential decay to rank relevance.

• Ncell network coverage map – signal strength decreases exponentially with distance.

Exam tip

• Know the formulas: $a_n$, $S_n$ for AP and GP, and the infinite GP sum condition $|r|<1$.
• Practice deriving nth terms from given first term and difference/ratio.
• Solve sum problems by substituting values directly into the closed‑form formulas; avoid summing term‑by‑term unless required.
• Check convergence: for infinite series, verify $|r|<1$ before applying the formula.
• Use diagrams: sketching the sequence or series on a number line or graph often clarifies the pattern and helps spot mistakes.