Sequence and Series: Sum of Finite Series – Types, Formulas & Applications

TAKEAWAYS:

• Understand arithmetic sequences (common difference) and geometric sequences (common ratio) to find their sums.
• Memorize sum formulas: $S_n=\frac{n}{2}[2a+(n-1)d]$ (arithmetic) and $S_n=a\frac{1-r^n}{1-r}$ (geometric).
• Recognize mixed sequences (alternating arithmetic/geometric) and use telescoping series for simplification.
• Apply sums to real problems like loan repayments, projectile motion, and population growth.
• Avoid common mistakes: misapplying formulas, ignoring convergence, or skipping step-by-step calculations.
• Practice NEB-style questions with multiple-choice, short-answer, and long-answer formats.

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1. What Are Sequences and Series?

A sequence is an ordered list of numbers following a rule. A series is the sum of its terms.

Types of Sequences

1. Arithmetic Sequence (AP)

   • Each term increases/decreases by a constant difference $d$.
   • Example: $3,7,11,15,\dots$ (here, $d=4$).

2. Geometric Sequence (GP)

   • Each term is multiplied by a constant ratio $r$.
   • Example: $2,6,18,54,\dots$ (here, $r=3$).

3. Mixed Sequences

   • Combine AP and GP rules (e.g., $1,4,9,16,\dots$ is quadratic, not AP/GP).

Visual: Arithmetic vs. Geometric Sequence

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2. Sum of Finite Arithmetic Series

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

• $a$ = first term,
• $d$ = common difference,
• $n$ = number of terms.

Derivation (Why This Formula Works)

1. Write the series forward and backward: $$S_n=a+(a+d)+(a+2d)+\cdots +l$$ $$S_n=l+(l-d)+(l-2d)+\cdots +a$$
2. Add both equations: $$2S_n=(a+l)+(a+l)+\cdots +(a+l)\text{(n times)}$$ $$2S_n=n(a+l)$$
3. The last term $l=a+(n-1)d$. Substitute: $$S_n=\frac{n}{2}[a+a+(n-1)d]=\frac{n}{2}[2a+(n-1)d]$$

Worked Example 1: Sum of an AP

Problem: Find the sum of the first 10 terms of the AP: $5,9,13,\dots$.

Solution:

• $a=5$, $d=4$, $n=10$.
• Plug into the formula: $$S_{10}=\frac{10}{2}[2(5)+(10-1)(4)]=5[10+36]=5\times 46=230$$

Answer: The sum is 230.

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3. Sum of Finite Geometric Series

The sum of the first $n$ terms of a GP is: $$S_n=a\frac{1-r^n}{1-r}\text{(if r=1)}$$ where:

• $a$ = first term,
• $r$ = common ratio,
• $n$ = number of terms.

Derivation (Why This Formula Works)

1. Write the series: $$S_n=a+ar+a{r}^2+\cdots +a{r}^{n-1}$$
2. Multiply by $r$: $$r{S}_n=ar+a{r}^2+a{r}^3+\cdots +a{r}^n$$
3. Subtract the two equations: $$S_n-r{S}_n=a-a{r}^n$$ $$S_n(1-r)=a(1-r^n)$$
4. Solve for $S_n$: $$S_n=a\frac{1-r^n}{1-r}$$

Worked Example 2: Sum of a GP

Problem: Find the sum of the first 5 terms of the GP: $3,6,12,\dots$.

Solution:

• $a=3$, $r=2$, $n=5$.
• Plug into the formula: $$S_5=3\frac{1-2^5}{1-2}=3\frac{1-32}{-1}=3\times 31=93$$

Answer: The sum is 93.

Special Case: Infinite GP ($|r|<1$)

If $|r|<1$, the series converges, and: $$S_{\infty }=\frac{a}{1-r}$$ Example: For $a=1$, $r=\frac{1}{2}$: $$S_{\infty }=\frac{1}{1-\frac{1}{2}}=2$$

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4. Mixed Sequences and Telescoping Series

Mixed Sequences

Some sequences combine AP and GP rules. Identify the pattern first! Example: Find the sum of $1+3+7+13+21$.

• Differences between terms: $+2,+4,+6,+8$ (AP of differences).
• This is a quadratic sequence. Use the general term formula: $$T_n=a{n}^2+bn+c$$ Solve for $a,b,c$ using given terms, then sum.

Telescoping Series

A series where most terms cancel out when expanded. Example: Sum ${\sum }_{k=1}^n\left(\frac{1}{k}-\frac{1}{k+1}\right)$.

• Expand: $$\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\cdots +\left(\frac{1}{n}-\frac{1}{n+1}\right)$$
• Most terms cancel, leaving: $$S_n=1-\frac{1}{n+1}=\frac{n}{n+1}$$

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5. Applications of Sum of Series

Real-World Problems

1. Finance:

   • Loan repayments: If you repay ₹1000 monthly with 10% interest, the total repayment is a GP sum.
   • Savings: Calculate total savings after $n$ years with monthly deposits.

2. Physics:

   • Projectile motion: Sum of distances traveled in each second (AP).
   • Wave interference: Sum of amplitudes (GP if decaying).

3. Engineering:

   • Signal processing: Sum of decaying signals (GP).
   • Structural analysis: Sum of forces (AP/GP).

Worked Example 3: Loan Repayment (GP Application)

Problem: A person takes a loan of ₹50,000 to be repaid in 5 equal annual installments of ₹12,000 each, with 10% interest. What is the total amount repaid?

Solution:

• Each installment is a GP where the first term $a=12,000$ and $r=1.10$ (10% interest).
• Total repayment $S_5=12,000\frac{1-1.10^5}{1-1.10}$.
• Calculate $1.10^5\approx 1.6105$.
• $S_5=12,000\frac{1-1.6105}{-0.10}=12,000\times 6.105=73,260$.

Answer: Total repayment is ₹73,260.

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6. Common Mistakes to Avoid

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7. NEB Exam Tips

What to Expect in Exams

1. Multiple-Choice Questions (MCQ):

   • Identify sequence type (AP/GP/mixed).
   • Calculate sums or missing terms.
   • Example:

     The sum of the first 8 terms of an AP is 120, and the 5th term is 15. Find the common difference. Options: (A) 2 (B) 3 (C) 4 (D) 5 Answer: (B) 3

2. Short-Answer Questions:

   • Derive sum formulas from first principles.
   • Solve for $n$ or $a$ given $S_n$.
   • Example:

     If the sum of the first $n$ terms of a GP is $2^n-1$, find the first term and common ratio.

3. Long-Answer Questions:

   • Real-world applications (loans, physics, etc.).
   • Prove a series telescopes or identify its type.
   • Example:

     A ball is dropped from a height of 10 m. It rebounds to 60% of its previous height each time. Find the total distance traveled when it hits the ground for the 4th time.

Marking Scheme Insights

• Step marks: Show all steps (e.g., identifying $a,d,r$, substituting into formulas).
• Formula marks: Write the correct formula before substituting values.
• Unit consistency: Ensure terms like "years," "meters," or "rupees" are consistent.

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8. Practice Questions (NEB Style)

Section A: Multiple Choice

1. The sum of the first 10 terms of the AP $2,5,8,\dots$ is: (A) 120 (B) 130 (C) 140 (D) 150 Answer: (B) 130

2. For a GP with $a=3$ and $r=2$, the sum of the first 4 terms is: (A) 45 (B) 48 (C) 42 (D) 36 Answer: (B) 48

Section B: Short Answer

3. Find the sum of the series $5+10+15+\cdots +50$. Answer: 150

4. The sum of the first $n$ terms of a GP is $2^n-1$. Find the common ratio. Answer: $r=2$

Section C: Long Answer

5. A farmer plants 100 trees in a row. Each subsequent row has 5 more trees than the previous. If there are 20 rows, find: (a) The number of trees in the 20th row. (b) The total number of trees planted. Answer: (a) 190 trees (b) 2,100 trees

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9. Summary Table: AP vs. GP Sum Formulas

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Exam Tip

• Memorize formulas but understand derivations—examiners test both!
• Draw diagrams for real-world problems (e.g., loan repayment tables, projectile motion graphs).
• Check units in word problems (e.g., years vs. months in finance).
• Practice mixed sequences—they often appear in higher-mark questions.
• Time management: Spend 1–2 minutes per MCQ, 5–10 minutes per short answer, and 15–20 minutes per long answer.

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A labelled diagram showing a GP with terms $a,ar,a{r}^2,\dots$ and arrows indicating common ratio $r$. (Image: Resident Mario (talk), CC BY-SA 3.0, via Wikimedia Commons)