Sequences & Series

A sequence is an ordered list of numbers following a pattern, and a series is the sum of terms of a sequence. These concepts appear everywhere -- from calculating loan payments to population growth models.

Arithmetic Progression (AP)

An AP has a common difference (d) between consecutive terms.

Example: 2, 5, 8, 11, 14, ... (d = 3)

nth term: an = a + (n - 1)d

where a is the first term and d is the common difference.

Sum of first n terms: Sn = n/2 [2a + (n - 1)d] = n/2 (a + l)

where l is the last term.

Worked Example (AP)

Find the sum of the first 20 terms of the AP: 3, 7, 11, 15, ...

Solution:

• a = 3, d = 7 - 3 = 4, n = 20
• Sn = n/2 [2a + (n-1)d]
• S20 = 20/2 [2(3) + 19(4)]
• S20 = 10 [6 + 76] = 10 x 82 = 820

Geometric Progression (GP)

A GP has a common ratio (r) between consecutive terms.

Example: 2, 6, 18, 54, ... (r = 3)

nth term: an = a r^(n-1)

Sum of first n terms: Sn = a(r^n - 1)/(r - 1) when r > 1

Sum of first n terms: Sn = a(1 - r^n)/(1 - r) when r < 1

Sum to infinity (|r| < 1): S_inf = a/(1 - r)

Worked Example (GP)

Find the sum of the first 5 terms of GP: 3, 6, 12, 24, ...

Solution:

• a = 3, r = 6/3 = 2, n = 5
• Sn = a(r^n - 1)/(r - 1) = 3(2⁵ - 1)/(2 - 1)
• S5 = 3(32 - 1)/1 = 3 x 31 = 93

Sum to Infinity Example

Find the sum of: 1 + 1/2 + 1/4 + 1/8 + ...

Solution:

• a = 1, r = 1/2 (|r| < 1, so it converges)
• S_inf = a/(1-r) = 1/(1 - 1/2) = 1/(1/2) = 2

Arithmetic Mean and Geometric Mean

• Arithmetic mean of a and b: AM = (a + b)/2
• Geometric mean of a and b: GM = sqrt(ab)
• Important property: AM >= GM (equality when a = b)

Real-World Applications

• AP: Monthly savings with fixed increments, equally spaced rungs of a ladder
• GP: Compound interest, population growth, radioactive decay, bacterial growth
• A Nepali student saves Rs. 100 in the first month and increases savings by Rs. 50 each month (AP). After 12 months, total savings = 12/2 [200 + 11(50)] = 6 x 750 = Rs. 4500.

Key Takeaways

• AP: constant difference between terms; an = a + (n-1)d
• GP: constant ratio between terms; an = ar^(n-1)
• Sum formulas are essential for solving series problems
• Infinite GP converges only when |r| < 1