CAT Quant Sequence and Series – Important Formulas and Concepts
Progression
Progression refers to a sequence of numbers arranged in a specific order, where each term follows a certain rule or pattern. There are three standard types of progression –
Arithmetic Progression
Geometric Progression
Harmonic Progression
Arithmetic Progression [AP]
An arithmetic progression is a sequence of numbers in which any number (except for the first) is more or less then the preceding number by a constant value.
The constant value is called the common difference, denoted by “d”.
The general form of an arithmetic progression is –> a, a+d, a+2d, a+3d ….. where a is the first term.
The sum or difference between any two consecutive terms is constant
Let “a” be the first term, “d” the common difference and “n” the number of terms in an AP, then the general form for the “nth term” is – $$T_n = a+(n-1)d$$
Sum of “n terms” in an AP is given by – $$S_n = \frac{n}{2}*[2a + (n-1)d] \\ \ \hspace{1cm} =\frac{n}{2}*[a + {a + (n-1)d}] \\ S_n = \frac{n}{2}*[First \ Term + Last \ Term]$$
Number of terms in an AP is given by – $$n = \frac{Last \ Term – First \ Term}{Common difference (d)} + 1$$
If a, b, c, d, … are in AP then and “k” is a constant term then –
a-k, b-k, c-k, ….. will also be in AP
ak, bk, ck, ….. will also be in AP
a/k, b/k, c/k, ….. will also be in AP
The Average of all terms in an AP is called their Arithmetic Mean. Arithmetic Mean of n terms in arithmetic progression is given by – $$AM = \frac{S_n}{n} = \frac{\frac{1}{2} *[2a + (n-1)d]}{n} \\ \hspace{1cm} = \frac{First \ Term + Last \ Term}{2}$$ Arithmetic Mean(AM) is the average of the first and last terms of the AP. It can also be obtained by taking the average of any two terms which are equidistant from two ends of the AP.
If three numbers are in AP, the middle term is called the arithmetic mean(AM). So if a, b, c are in AP then $$AM = b = \frac{a+c}{2}$$
If two numbers a and b are in AP, then their arithmetic mean(AM) is – $$AM = \frac{(a+b)}{2}$$
If three or more numbers are in AP, we can represent them as –
for 3 Numbers –> (a-d), a and (a+d).
for 4 Numbers –> (a-3d), (a-d), (a+d) and (a+3d).
for 5 Numbers –> (a-2d), (a-d), a, (a+d) and (a+2d).
Geometric Progression [GP]
A sequence is said to be in Geometric progression, if the ratio of any number (except for the first) to the preceding one is the same.
The constant ratio is called the Common Ratio, denoted by “r”.
Any term of a geometric progression can be obtained by multiplying the preceding number by the common ratio.
The general form of a geometric progression is –> a, ar, ar2, ar3, ….. where a is the first term.
Let “a” be the first term, “r” the common difference and “n” the number of terms in a GP, then the general form for the “nth term” is – $$T_n = ar^{(n-1)}$$
Sum of “n terms” in a GP is given by – $$S_n = \frac{a*(1-r^n)}{1-r} \ or \ \frac{a(r^n – 1)}{r-1} \\ S_n = \frac{r*(Last \ Term) – First \ Term}{r-1}$$
If a, b, c, d, … are in GP then and “k” is a constant term then –
ak, bk, ck, ….. will also be in GP
a/k, b/k, c/k, ….. will also be in GP
If n terms a1, a2, a3, ….. an are in GP, then the Geometric Mean (GM) of these n terms is given by – $$GM = \sqrt[n]{a_1*a_2*a_3* ….. a_n} $$
If three terms are in GP, the middle term is called the geometric mean(GM). So if a, b, c are in GP then $$GM = b = \sqrt{ac}$$
If two terms a and b are in GP, then their geometric mean(GM) is given by – $$GM = \sqrt{ab}$$
If three or more numbers are in GP, we can represent them as –
for 3 Terms –> (a/r), a and (ar).
for 4 Terms –> (a/r3), (a/r), (ar) and (ar3). In this case r2 is the common ratio.
NOTE – For any two unequal positive numbers a and b, their Arithmetic Mean is always greater than their Geometric Mean – $$\frac{(a+b)}{2} > \sqrt{ab} \quad \ \longrightarrow \quad (a+b) > 2\sqrt{ab}$$
Infinite Geometric Progression
IF -1 < r < +1 or |r| < 1, the sum of a geometric progression does not increase infinitely; it converges to a particular value. Such a GP is referred to as an infinite geometric progression.
The Sum of an Infinite Geometric Progression is given by – $$S_{\infty} = \frac{a}{1-r}$$
Harmonic Progression [HP]
If the reciprocals of the terms of a sequence are in AP, the sequence is said to be in Harmonic Progression.
The general form of a Harmonic Progression is –> \( \frac{1}{a} , \frac{1}{a+d} , \frac{1}{a+2d} ……… \), where \( \frac{1}{a} \) is the first term.
If a, b, c, d,.. are the given numbers in H.P then the Harmonic mean of – $$”n \ terms” = \frac{Number \ of \ terms}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}……… }$$
If three terms are in HP, the middle term is called the harmonic mean(HM). So if a, b, c are in HP then b is said to be the Harmonic Mean of a and c.
For two numbers a and b, their Harmonic Mean(HM) is given by – $$HM = \frac{2ab}{a+b}$$
For any two positive numbers a and b, $$AM \geqslant \ GM \geqslant \ HM $$ $$GM = \sqrt{Am * HM}$$
Arithmetic Geometric Series
A series is considered an arithmetic-geometric series if each of its terms is obtained by the product of the corresponding terms of an arithmetic progression (AP) and geometric progression (GP).
The general form of Arithmetic Geometric Series is –> a,(a + 2d)r, (a + 2d)r2, …..
The Sum of “n terms” of an AGP series is given by – $$S_{\infty} = \frac{a – [a + (n – 1)d] r^n}{1-r} + \frac{dr (1 – r^{(n – 1)})}{(1 – r)^2} \hspace{1cm} [r \neq 1]$$
Sum of Infinite terms of AGP series is given by – $$S_{\infty} = \frac{a}{1-r} + \frac{dr}{(1 – r)^2} \hspace{1cm} [|r| < 1]$$
Some Important Series
Sum of the first n natural numbers – $$\sum n = \frac{n(n+1)}{2}$$
Sum of squares of first n natural numbers – $$\sum n^2 = \frac{n(n+1)(2n +1)}{6}$$
Sum of cubes of first n natural numbers – $$\sum n^3 = {\LARGE[} \frac{n(n+1)}{2} {\LARGE]}^2= \frac{n^2(n+1)^2}{4} = {\LARGE[} \sum n {\LARGE]}^2$$
The sum of first ‘n’ odd natural numbers = n2
The sum of first ‘n’ even natural numbers = n(n+1)
In any series, if the sum of first n terms is given by 𝑆n, then the “nth term” is given by –> \(T_n = S_n \ – \ S_{n-1}\)
