Progressions: Concepts, Formulas, and Clear Examples

AP has constant difference $d$; GP has constant ratio $r$. Sum formulas turn long adds into one multiplication—essential under time pressure.

Campus items: find $n$th term, sum of $n$ terms, infinite GP when $|r|<1$, and insert means between two numbers.

For AP, average of first and last equals middle term in symmetric pairs.

AP terms and sum

nth term AP

$$a_n = a + (n-1)d$$

Sum n terms AP

$$S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + l)$$

Example — Sum 1..100

$S=\frac{100\cdot101}{2}=5050$.

GP terms and sum

nth term GP

$$a_n = ar^{n-1}$$

Sum n terms GP

$$S_n = a\frac{r^n-1}{r-1},\ r\neq1$$

Infinite GP

$$S_\infty = \frac{a}{1-r},\ |r|<1$$

Insert means

• $m$ AMs between $a,b$: common difference $d=\frac{b-a}{m+1}$.
• GM between $a,b$: $\sqrt{ab}$ (single).

Properties AP

• If $m+n=2k$, then $a_m+a_n=2a_k$.
• Sum of first $n$ odd numbers = $n^2$.

Properties GP

• Product of equidistant terms constant in finite GP patterns.
• Logs turn GP into AP.

Hybrid MCQs

• Identify AP/GP from term differences or ratios of consecutive terms.

Step-by-step method for exam questions

1. Check consecutive differences (AP) or ratios (GP).
2. Write $a,d$ or $a,r$ from given terms.
3. Form equation for unknown $n$ or sum.
4. Apply closed-form sum; avoid term-by-term add.
5. For infinite GP verify $|r|<1$.
6. Pick option.

Formula cheat sheet

Frequently asked questions

AP with zero d?

Constant sequence; sum = $na$.

GP r=1?

Sum = $na$; formula division by zero—handle separately.

Three numbers in AP?

Often use $a-d, a, a+d$.

Find n from sum equation?

Quadratic in n common; take positive integer root.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.