Unit 6: Sequence and Series
1. Arithmetic Progression (A.P.)
A sequence is an Arithmetic Progression if the difference between any two consecutive terms is constant. This constant difference is called the Common Difference (d).
General Term (nth term): an = a + (n - 1)d
Sum of n terms (Sn): Sn = (n) / (2)[2a + (n - 1)d] OR Sn = (n) / (2)[a + l]
Where a = first term, d = common difference, n = number of terms, and l = last term.
2. Geometric Progression (G.P.)
A sequence is a Geometric Progression if the ratio of any two consecutive terms is constant. This constant ratio is called the Common Ratio (r).
General Term (nth term): an = arn-1
Sum of n terms (Sn):
If r ≠ 1, Sn = (a(rn - 1)) / (r - 1) (for r > 1) OR Sn = (a(1 - rn)) / (1 - r) (for r < 1)
Sum of Infinite G.P.: If |r| < 1, the sum of an infinite G.P. is S∞ = (a) / (1 - r).
3. Arithmetic Mean (A.M.) and Geometric Mean (G.M.)
Means are values inserted between two numbers to form a progression.
- Single A.M. between a and b: A = (a + b) / (2)
- Single G.M. between a and b: G = √(ab)
Property: For any two positive real numbers a and b, A.M. ≥q G.M.
4. Summation of Special Series
These standard results are used to find the sum of series that are not directly in A.P. or G.P.
Sum of first n natural numbers: ∑ n = (n(n + 1)) / (2)
Sum of squares: ∑ n2 = (n(n + 1)(2n + 1)) / (6)
Sum of cubes: ∑ n3 = ≤ft[ (n(n + 1)) / (2) ]2
5. Exam Focus Enhancements
Exam Tips
- Selection of Terms: If you need to assume 3 terms in A.P., use (a-d), a, (a+d). For G.P., use (a) / (r), a, ar. This simplifies calculations significantly.
- Infinite G.P.: This only works if |r| < 1. If r ≥q 1, the sum is infinite.
- Method of Difference: For complex series, find the nth term (Tn) first, then apply the ∑ operator to find Sn.
Common Mistakes
- Confusing d and r: Ensure you subtract for A.P. (a2 - a1) and divide for G.P. (a2 / a1).
- n vs an: n is the position (index), while an is the value at that position.
- Sum of Squares Formula: Forgetting the (2n+1) part of the ∑ n2 formula is a very common error.
Frequently Asked Questions
Q: Can d or r be negative?
A: Yes. A negative d means a decreasing A.P. A negative r means the signs of the terms will alternate.
Q: What is the relationship between A.M. and G.M.?
A: A.M. ≥q G.M.. They are equal only if the numbers a and b are equal.