MH Board · Chapter

Sequences and Series

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\sum r^3 = \left(\frac{n(n+1)}{2}\right)^2

the first

n

Start

A.P. and its nth termAn A.P. adds the same

d

each time:

t_n = a + (n-1)d

Locked

Sum of n terms of an A.P.

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

: pair the first and last termsLocked

Three or four numbers in A.P.Take

a - d, a, a + d

so the sum fixes

a

; for consecutive terms

2b = a + c

Locked

Spotting a G.P.A G.P. multiplies by the same $r$ each time: $r = \frac{t_{n+1}}{t_n}$ , not a differenceLocked
The nth term of a G.P. $t_n = ar^{n-1}$ : find a term, which term, or $a$ and $r$
Three or four numbers in G.P.Take $\frac{a}{r}, a, ar$ (or $\frac{a}{r^3}, \frac{a}{r}, ar, ar^3$ )
Sum of n terms of a G.P. $S_n = \frac{a(r^n - 1)}{r - 1}$
Sums like 5 + 55 + 555Write $55\ldots5$ as $\frac{5}{9}(10^k - 1)$ ,
The nth term from the sum $t_n = S_n - S_{n-1}$
Sum to infinityWhen $|r| < 1$ , $S_\infty = \frac{a}{1 - r}$ ;
Recurring decimals as fractions $0.\overline{36} = \frac{36}{100} + \frac{36}{100^2} + \ldots = \frac{36}{99}$

Harmonic progressionTake reciprocals: they form an A.P.; work there, then flip backLocked
A.M., G.M. and H.M. $A = \frac{a+b}{2}$ , $G = \sqrt{ab}$ , $H = \frac{2ab}{a+b}$ ; $G^2 = AH$ and $A \ge G \ge H$ Locked
Inserting means $k$ numbers between $a$ and $b$ make $k + 2$ terms: find $d$

Arithmetico-geometric seriesMultiply $S$ by $r$ , subtract, and the rest is a G.P.Locked
Sums of r, r² and r³ $\sum r = \frac{n(n+1)}{2}$ , $\sum r^2 = \frac{n(n+1)(2n+1)}{6}$ , $\sum r^3 = \left(\frac{n(n+1)}{2}\right)^2$ Start
Summing a series with ΣWrite the $r$ th term, expand it, and sum each power of $r$ separatelyLocked

from two terms

Locked

so the product gives

a

at once

Locked

for

r \ne 1

;

S_n = na

when

r = 1

Locked

then sum a G.P. and a constant

Locked

for

n \ge 2

,

and

t_1 = S_1

Locked

otherwise there is no sum

Locked

or

r

from the last term

Locked