Sequence and Series: Types, Formulas & Applications

TAKEAWAYS:

• A sequence is a list of numbers in a specific order (e.g., 2, 4, 6, 8…), while a series is the sum of its terms (e.g., 2 + 4 + 6 + 8…).
• Arithmetic sequences grow by a constant difference (e.g., 3, 7, 11…), and their sum uses the formula $S_n=\frac{n}{2}[2a+(n-1)d]$.
• Geometric sequences multiply by a common ratio (e.g., 5, 15, 45…), and their sum is $S_n=a\frac{1-r^n}{1-r}$ (if $r\ne 1$).
• Infinite geometric series converge only if $|r|<1$, with sum $S=\frac{a}{1-r}$.
• Applications include calculating loan installments, compound interest, and depreciation of assets.
• Always check if a sequence is arithmetic, geometric, or neither before applying formulas.

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1. What is a Sequence?

A sequence is an ordered list of numbers where each number is called a term. Sequences can be finite (limited terms) or infinite (unlimited terms).

Types of Sequences

1. Finite Sequence: Has a fixed number of terms (e.g., 1, 3, 5, 7).
2. Infinite Sequence: Continues forever (e.g., 2, 4, 6, 8, …).

How to Identify a Sequence?

Look for a pattern in the terms. Common patterns include:

• Arithmetic: Adding/subtracting a fixed number (e.g., 5, 8, 11… → +3 each time).
• Geometric: Multiplying/dividing by a fixed number (e.g., 3, 6, 12… → ×2 each time).
• Other patterns: Squares, cubes, factorials, or mixed operations.

Example 1: Identifying a Sequence

Question: Is the sequence 2, 5, 10, 17… arithmetic, geometric, or neither? Solution:

• Check differences: 5–2=3, 10–5=5, 17–10=7 → Not arithmetic (difference changes).
• Check ratios: 5/2=2.5, 10/5=2, 17/10=1.7 → Not geometric (ratio changes).
• Answer: Neither. The pattern is $n^2+1$ (for $n=1,2,3\dots$).

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2. Arithmetic Sequence (AP)

An arithmetic sequence (AP) is a sequence where each term increases or decreases by a constant difference ($d$).

General Form of AP

$$a,a+d,a+2d,a+3d,\dots$$

• $a$ = first term
• $d$ = common difference

nth Term of AP

The $n$-th term ($T_n$) is given by: $$T_n=a+(n-1)d$$

Example 2: Finding the 10th Term of an AP

Question: Find the 10th term of the AP: 7, 12, 17, 22… Solution:

• $a=7$
• $d=12-7=5$
• $n=10$ $$T_{10}=7+(10-1)\times 5=7+45=52$$ Answer: The 10th term is 52.

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Sum of First $n$ Terms of AP ($S_n$)

$$S_n=\frac{n}{2}[2a+(n-1)d]$$ Alternative formula (if last term $l$ is known): $$S_n=\frac{n}{2}(a+l)$$

Example 3: Sum of an AP

Question: Find the sum of the first 12 terms of the AP: 3, 8, 13, 18… Solution:

• $a=3$
• $d=8-3=5$
• $n=12$ $$S_{12}=\frac{12}{2}[2\times 3+(12-1)\times 5]=6[6+55]=6\times 61=366$$ Answer: The sum is 366.

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3. Geometric Sequence (GP)

A geometric sequence (GP) is a sequence where each term is multiplied by a common ratio ($r$).

General Form of GP

$$a,ar,a{r}^2,a{r}^3,\dots$$

• $a$ = first term
• $r$ = common ratio

nth Term of GP

$$T_n=a\times r^{n-1}$$

Example 4: Finding the 6th Term of a GP

Question: Find the 6th term of the GP: 4, 12, 36, 108… Solution:

• $a=4$
• $r=12/4=3$
• $n=6$ $$T_6=4\times 3^{6-1}=4\times 3^5=4\times 243=972$$ Answer: The 6th term is 972.

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Sum of First $n$ Terms of GP ($S_n$)

$$S_n=a\frac{1-r^n}{1-r}\text{(if r=1)}$$ If $r=1$, the GP becomes a constant sequence: $$S_n=n\times a$$

Example 5: Sum of a GP

Question: Find the sum of the first 8 terms of the GP: 5, 15, 45, 135… Solution:

• $a=5$
• $r=15/5=3$
• $n=8$ $$S_8=5\frac{1-3^8}{1-3}=5\frac{1-6561}{-2}=5\times \frac{-6560}{-2}=5\times 3280=16400$$ Answer: The sum is 16,400.

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4. Infinite Geometric Series

An infinite geometric series has terms that go on forever. It converges (has a finite sum) only if $|r|<1$.

Sum of Infinite GP ($S$)

$$S=\frac{a}{1-r}\text{(if ∣r∣<1)}$$

Example 6: Sum of an Infinite GP

Question: Find the sum of the infinite series: 100 + 50 + 25 + 12.5 + … Solution:

• $a=100$
• $r=50/100=0.5$ (since $|0.5|<1$, it converges) $$S=\frac{100}{1-0.5}=\frac{100}{0.5}=200$$ Answer: The sum is 200.

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5. Applications of Sequences and Series in Business

Sequences and series are used in:

1. Loan Repayments: Calculating equal installments (arithmetic sequence).
2. Depreciation: Reducing asset value over time (geometric sequence).
3. Investment Growth: Compound interest (geometric series).
4. Profit Analysis: Summing sales over periods (arithmetic series).

Example 7: Loan Repayment (Arithmetic Sequence)

Question: A loan of Rs. 10,000 is repaid in 5 equal installments of Rs. 2,500 each. What is the total interest paid? Solution:

• Total repayment = $5\times 2500=12,500$
• Interest = Total repayment – Principal = $12,500-10,000=2,500$ Answer: Total interest paid is Rs. 2,500.

Example 8: Depreciation (Geometric Sequence)

Question: A machine depreciates at 10% annually. If its initial value is Rs. 50,000, what is its value after 3 years? Solution:

• Depreciation rate ($r$) = 10% = 0.10
• Value after 3 years = $50,000\times (1-0.10{)}^3=50,000\times 0.729=36,450$ Answer: The machine’s value after 3 years is Rs. 36,450.

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6. Comparison: Arithmetic vs. Geometric Sequences

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7. Common Mistakes to Avoid

1. Assuming all sequences are AP or GP: Always check the pattern first.
2. Forgetting $n-1$ in the nth term formula: $T_n=a+(n-1)d$, not $a+nd$.
3. Incorrectly applying the sum formula for GP: Ensure $r\ne 1$ and $|r|<1$ for infinite series.
4. Miscounting terms: Start counting from $n=1$, not $n=0$.

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8. NEB-Style Practice Questions

Short Answer Questions (SAQ)

1. Define an arithmetic sequence. Give an example.
2. Write the formula for the sum of the first $n$ terms of a GP.
3. When does an infinite geometric series converge? Give the sum formula.
4. Find the 7th term of the AP: 10, 15, 20, 25…
5. Calculate the sum of the first 6 terms of the GP: 2, 6, 18, 54…

Long Answer Questions (LAQ)

6. A company’s profit increases by Rs. 5,000 every year. If the profit in the first year was Rs. 20,000, find:
   • The profit in the 10th year.
   • The total profit over 10 years.
7. A machine loses 5% of its value every year. If its initial value is Rs. 80,000, find:
   • Its value after 4 years.
   • The total depreciation over 4 years.
8. An infinite series has the first term $a=4$ and common ratio $r=0.5$. Find its sum.
9. The sum of the first 5 terms of an AP is 75, and the first term is 11. Find the common difference.
10. A loan of Rs. 50,000 is to be repaid in 4 equal installments. If the total interest is Rs. 5,000, find the annual installment amount.

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Exam Tip

1. Always identify the type of sequence (AP or GP) before applying formulas.
2. Double-check calculations, especially for $r^n$ in GPs.
3. Memorize key formulas:
   • AP nth term: $T_n=a+(n-1)d$
   • AP sum: $S_n=\frac{n}{2}[2a+(n-1)d]$
   • GP nth term: $T_n=a\times r^{n-1}$
   • GP sum: $S_n=a\frac{1-r^n}{1-r}$
4. For infinite series, ensure $|r|<1$ before using $S=\frac{a}{1-r}$.
5. Business applications often involve real-world scenarios like loans, depreciation, or interest—relate theory to practice.

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Good luck with your NEB exam preparation! 🚀