Geometric Progression (GP) — Complete Formula

Basic Definition of GP

Concept	Formula	Symbols Meaning	Key Notes / Tricks
General Form	[a, ar, ar^2, ar^3, …]	[a] = first term, [r] = common ratio	Ratio remains constant ⭐
Common Ratio	[r = \dfrac{a_2}{a_1}]	[a_1, a_2] = consecutive terms	Also [r = \dfrac{a_n}{a_{n-1}}] ⭐

💡 Memory Hint:
GP → multiply by r each time

Concept	Formula	Symbols Meaning	Key Notes / Tricks
nth Term	[a_n = a r^{n-1}]	[a_n] = nth term, [n] = term number	Most important ⭐
Alternate Form	[a_n = a_m r^{n-m}]	[a_m] = mth term	Useful in JEE ⭐

💡 Memory Hint:
Each term = previous × [r]

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Sum (r ≠ 1)	[S_n = a \dfrac{r^n – 1}{r – 1}]	[S_n] = sum of n terms	Standard form ⭐
Alternate Form	[S_n = a \dfrac{1 – r^n}{1 – r}]	—	Use when [r < 1] ⭐
Special Case	[S_n = na] (if [r = 1])	—	All terms equal

💡 Memory Hint:
Two forms → choose based on ease of simplification

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Infinite Sum	[S_\infty = \dfrac{a}{1 – r}]		Valid when [-1 < r < 1]

💡 Memory Hint:
Converges only if |r| < 1

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Last Term	[l = a r^{n-1}]	[l] = last term	Same as nth term ⭐

Concept	Formula	Symbols Meaning	Key Notes / Tricks
From nth Term	[n = 1 + \dfrac{\log(a_n/a)}{\log r}]	—	Log-based ⭐
From Sum	Solve equation in [r^n]	—	Often logarithmic ⭐

💡 Memory Hint:
GP problems often → logarithms

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Single GM	[GM = \sqrt{ab}]	[a, b] = numbers	Important ⭐
Relation with AM	[AM \ge GM]	—	Equality when [a = b] ⭐

💡 Memory Hint:
GM is always ≤ AM

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Common Ratio	[r = \left(\dfrac{b}{a}\right)^{\dfrac{1}{n+1}}]	[n] = number of means	Important ⭐
Terms	[a r, a r^2, …]	—	Construct GP

💡 Memory Hint:
Divide ratio into equal multiplicative steps

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Sum of Powers	[1 + r + r^2 + … + r^{n-1} = \dfrac{r^n – 1}{r – 1}]	—	Base identity ⭐
Infinite Series	[1 + r + r^2 + … = \dfrac{1}{1 – r}]	[r] = Common ratio

💡 Memory Hint:
GP sum builds from geometric growth

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