Progressions (AP, GP..)

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Progressions (AP, GP..) — Quant Aptitude Questions & Formulas

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Progressions (AP, GP, HP)

Arithmetic Progression (AP)

$n$ -th term: $T_n = a + (n-1)d$
Sum: $S_n = \dfrac{n}{2}\big[2a + (n-1)d\big] = \dfrac{n}{2}(a + l)$
Arithmetic mean of $a, b$ : $\dfrac{a+b}{2}$

Tricks: if the sum of the first $p$ terms equals the sum of the first $q$ , then $S_{p+q} = 0$ . If $T_p = q$ and , then . Assume 3 terms as ; 4 terms as .

Geometric Progression (GP)

$n$ -th term: $T_n = a\,r^{\,n-1}$
Sum: $S_{n}$ (for ) or (for )

Harmonic Progression (HP)

$a, b, c$ are in HP iff $\frac1a, \frac1b, \frac1c$ are in AP. Harmonic mean of : (the same formula as equal-distance average speed!).

AM – GM – HM

Order: $AM \ge GM \ge HM$ (equal only when all values are identical).
$AM \times HM = GM^{2}$ .

Special Series

$\sum n = \dfrac{n(n+1)}{2}$
$\sum n^{2} = \dfrac{n(n+1)(2n+1)}{6}$

=

\frac{a (r^{n} - 1)}{r - 1}

S_n = \dfrac{a(r^{n}-1)}{r-1}