Sequences & Series – AP, GP, Sums, Convergence, Business Applications

1. Basic Definitions

The difference $d$ is added each step; the ratio $r$ multiplies each step.

2. Arithmetic Progression (AP)

2.1 nth‑term formula

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

2.2 Sum of first $N$ terms

$$S_N=\frac{N}{2}(2a_1+(N-1)d)=\frac{N}{2}(a_1+a_N)$$

2.3 Worked Example – Finding a term

Problem: Which term of the arithmetic series $6+10+14+\dots$ equals $62$?

Solution steps

1. Identify $a_1=6$ and $d=4$.
2. Set $a_n=62$ and solve for $n$:

$$\begin{gathered} 62=6+(n-1)4 \\ 56=4(n-1) \\ n-1=14 \\ n=15 \end{gathered}$$

Answer: The 15‑th term is $62$.

2.4 Business Use – Linear Cost Growth

A manufacturing firm’s raw‑material cost rises by NPR 500 per month. If the cost in January is NPR 8,000, the cost in month $n$ is an AP:

$$C_n=8000+(n-1)\times 500$$

The total cost for the first 12 months is obtained with the AP sum formula.

3. Geometric Progression (GP)

3.1 nth‑term formula

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

3.2 Sum of first $N$ terms

$$S_N=a_1\frac{1-r^N}{1-r},r\ne 1$$

3.3 Infinite geometric series

If $|r|<1$,

$$S_{\infty }=\frac{a_1}{1-r}$$

Otherwise the series diverges (no finite sum).

3.4 Worked Example – Sum of a GP

Problem: Find the sum of the geometric series $1+3+9+\dots$ to 8 terms.

Solution steps

1. Identify $a_1=1$, $r=3$, $N=8$.
2. Apply the finite‑sum formula:

$$S_8=1\frac{1-3^8}{1-3}=\frac{1-6561}{-2}=\frac{-6560}{-2}=3280$$

Answer: The sum of the first eight terms is 3,280.

3.5 Business Use – Compound Interest

A bank offers 12 % annual interest compounded yearly. The future value after $n$ years of an initial deposit $P$ is a GP:

$$FV=P(1+0.12{)}^n$$

The sum of a series of equal annual deposits (an annuity) also uses the GP sum formula.

4. Comparison of AP and GP

flowchart LR
    A["AP: add constant d"] --> B["Linear growth"]
    A --> C["Sum formula involves N/2"]
    D["GP: multiply by constant r"] --> E["Exponential growth"]
    D --> F["Sum formula involves (1‑r^N)/(1‑r)"]

5. Convergence & Divergence

• Convergent series: Partial sums approach a finite limit.
• Divergent series: Partial sums grow without bound or oscillate.

Key test for GP: $|r|<1$ ⇒ convergent; otherwise divergent.

Example – Present Value of Perpetual Cash Flow

A telecom company receives NPR 1,000 monthly forever, discount rate 5 % per month. The present value is an infinite GP with $a_1=1000$, $r=1/(1+0.05)=0.95238$.

$$PV=\frac{1000}{1-0.95238}=\frac{1000}{0.04762}\approx 21,000\text{ NPR}$$

This calculation underlies pricing of long‑term service contracts for Ncell.

6. Applications in Business

7. Worked Business Example – Daraz Order Queue

Daraz processes orders in a first‑in‑first‑out queue. Suppose each day the number of new orders grows by 20 % (GP) and the warehouse can ship a fixed 500 orders per day (AP of shipments).

• Orders on day $n$: $O_n=200\times 1.20^{n-1}$
• Cumulative orders after 5 days:

$$S_5=200\frac{1-1.20^5}{1-1.20}=200\frac{1-2.48832}{-0.20}=200\frac{-1.48832}{-0.20}=200\times 7.4416=1,488.32\approx 1,488\text{ orders}$$

• Orders shipped in 5 days: $5\times 500=2,500$

Since shipped capacity exceeds cumulative orders, no backlog occurs. If the growth rate rises to 40 %, the same calculation shows a backlog, prompting Daraz to increase daily shipping capacity.

8. Real‑World Connections

8.1 eSewa – Transaction Fee Structure

eSewa charges a fixed fee plus a 1 % variable fee on each transaction. The total fee for $n$ transactions of equal amount $A$ forms an AP:

$${\text{Fee}}_n=n\cdot (\text{fixed})+A\cdot 0.01\cdot n$$

Understanding AP sums helps merchants forecast monthly e‑payment costs.

8.2 Ncell – Monthly Data Bundle Growth

Ncell offers a data bundle that increases by 10 % each month for loyal customers. The bundle size follows a GP, enabling the company to model network load and price tiers.

8.3 NEPSE – Index Compounding

If the NEPSE index grows at an average annual rate of 8 %, the projected index after 5 years is a GP:

$${\text{Index}}_5={\text{Index}}_0(1.08{)}^5$$

Investors use this to estimate long‑term portfolio growth.

9. Summary of Formulas

**Arithmetic Sequence**
- nth term: \(a_n = a_1 + (n-1)d\)
- Sum of N terms: \(S_N = \frac{N}{2}(2a_1+(N-1)d)\)

**Geometric Sequence**
- nth term: \(a_n = a_1 r^{\,n-1}\)
- Sum of N terms: \(S_N = a_1\frac{1-r^{N}}{1-r}\)  ( \(r\neq1\) )
- Infinite sum (|r|<1): \(S_{\infty}= \frac{a_1}{1-r}\)

10. Common Mistakes to Avoid

Exam tip

• Identify the pattern quickly: constant addition → AP; constant multiplication → GP.
• Write the generic formula before plugging numbers; this reduces algebraic slips.
• For sum questions, decide whether the series is finite or infinite; apply the appropriate sum formula.
• Mark the condition $|r|<1$ when an infinite GP is asked – if not satisfied, state “diverges”.
• Use the compact AP sum form $\frac{N}{2}(a_1+a_N)$ when both first and last terms are known; it saves time.
• In multi‑step business problems (e.g., Daraz queue), draw a quick number line or bar chart to visualise growth; the visual often reveals whether capacity is sufficient.