Class 11 Maths • Chapter 08 • Comprehensive Interactive Notes
Sequence: An arrangement of numbers in a definite order according to some rule.
Series: If \( a_1, a_2, ..., a_n \) is a sequence, then the sum \( a_1 + a_2 + ... + a_n \) is called the series associated with it.
Sequence where difference between terms is constant.
| Type | \( n^{th} \) Term (\( a_n \)) | Sum (\( S_n \)) |
|---|---|---|
| AP | \( a + (n-1)d \) | \( \frac{n}{2}[2a + (n-1)d] \) |
| GP | \( ar^{n-1} \) | \( \frac{a(r^n - 1)}{r - 1} \) |
For two positive numbers \( a \) and \( b \):
Enter two positive numbers.
Sum of first \( n \) natural numbers and powers.
| Series | Sum Formula |
|---|---|
| \( \sum n = 1+2+...+n \) | \( \frac{n(n+1)}{2} \) |
| \( \sum n^2 = 1^2+2^2+...+n^2 \) | \( \frac{n(n+1)(2n+1)}{6} \) |
| \( \sum n^3 = 1^3+2^3+...+n^3 \) | \( [\frac{n(n+1)}{2}]^2 \) |
Calculate sums for specific n.
Consider the series: \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... \)
Common Ratio \( |r| < 1 \). Sum to infinity \( S_\infty=\frac{a}{1-r} \).
The infinite bars will never exceed total length 2. (Sum = 2)
Example 1 (AP):
The monthly salary of a worker increases by ₹500 every year. If his first salary is ₹15,000, find his salary after 10 years.
Example 2 (GP):
A ball rebounds to half its height every time. If initial height is 10 m, find total distance travelled.
Exam Insight: These questions test your ability to **identify a, d/r, n** correctly.
1. The 10th term of AP: 2, 7, 12... is:
2. If AM = 10 and GM = 8, the numbers are:
3. Sum of infinite GP exists only when:
4. Sum of first n natural numbers is:
5. Which sequence has a constant ratio?
Self-Check: