“Difference or ratio”
Spotting the progression
Subtract consecutive terms: constant means AP. Divide them: constant means GP. Do both on the first three terms before anything else.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Sequences and Series formulas
Arithmetic and geometric progressions are the most formula-driven chapter in Class 11, and also the most mechanical. Identify which progression you have, write the nth term, and nearly every question answers itself.
15 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Arithmetic progression
| What | Formula | Symbols and notes |
|---|---|---|
| nth term | an = a + (n − 1)d | d is the common difference |
| Sum of n terms | Sn = (n/2)[2a + (n − 1)d] | also (n/2)(a + l), where l is the last term |
| Arithmetic mean | AM = (a + b)/2 | the middle term of three in AP |
| n terms in AP | take them as a − d, a, a + d | symmetric choice keeps the algebra short |
| Sum of first n natural numbers | n(n + 1)/2 | a special case with a = d = 1 |
Geometric progression
| What | Formula | Symbols and notes |
|---|---|---|
| nth term | an = arn−1 | r is the common ratio |
| Sum of n terms | Sn = a(rn − 1)/(r − 1) | for r ≠ 1 |
| Sum to infinity | S∞ = a/(1 − r) | only when |r| < 1 |
| Geometric mean | GM = √(ab) | for two positive numbers |
| Three terms in GP | take them as a/r, a, ar | again for symmetry |
Means and standard sums
| What | Formula | Symbols and notes |
|---|---|---|
| AM and GM inequality | AM ≥ GM | equal only when the numbers are equal |
| Harmonic mean | HM = 2ab/(a + b) | for two numbers |
| Relation | GM² = AM × HM | and AM ≥ GM ≥ HM |
| Sum of squares | Σn² = n(n + 1)(2n + 1)/6 | for the first n natural numbers |
| Sum of cubes | Σn³ = [n(n + 1)/2]² | the square of the sum of the first n numbers |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Infinity needs a fraction”
Sum to infinity of a GP
S∞ = a/(1 − r) exists only when |r| < 1. If the ratio is 2, the sum simply grows without limit.
“AM beats GM beats HM”
The order of the three means
For positive unequal numbers, AM > GM > HM, and all three are equal only when the numbers are the same.
Where marks go
The mistakes that cost the most.
- Using n as the number of terms when counting from a term other than the first.
- Applying the infinite GP sum when |r| ≥ 1, where no finite sum exists.
- Using the AP sum formula with the last term unknown. Use the 2a + (n − 1)d version instead.
- Forgetting that the GP sum formula divides by r − 1, so r = 1 needs the separate case S = na.
- Taking the geometric mean of two numbers of opposite sign, which has no real value.
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