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Parin Sir > Quantitative Aptitude (Maths) > Fundamental (Numbers) > Progressions > Progressions

Progressions

May 28, 2020
Category: Progressions

■Sequence : A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a sequence.

The number occurring at the n-th place of a sequence is called its n-th term.

■Progression : If the terms of a sequence increase or decrease continuously, then such a sequence is called a progression.

■Arithmetical Progression (A.P.) : If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is known as common difference of the A.P.

An A.P. with first term = a & common difference = d is given by :

a, (a + d), (a + 2d), (a + 3d), …

The n-th term of the A.P. with 1^st term = a & common difference = d is given by :

T_n = a + (n – 1)d.

The sum of the n terms of an A.P. is given by :

S_n = $\frac{n}{2}$ [2a + (n – 1) d]

= $\frac{n}{2}$ ( first term + last term) = $\frac{n}{2}$ ( a + l ) : Where, l = last term

e.g. (i) How many numbers between 16 and 200 are divisible by 7 ?

Soln. The required numbers are 21, 28, 35, …, 196.

This is clearly and A.P. in which a = 21 & d = 7.

Let the number of terms of this A.P. be n.

Then, T_n = 196 $\Rightarrow$ a + (n – 1)d = 196, i.e. 21 + (n – 1) 7 = 196

$\Rightarrow$ n = 26.

Required number of numbers = 26.

Note : Another method to solve the above example is :

For the given sequence, $T_{n}$ = 7n + 14

$\therefore$ 7n + 14 = 196

$\therefore$ n = 26

e.g. (ii) Find the sum of first 50 natural numbers.

Soln. We have to find : 1 + 2 + 3 +…+ 50

This is an A.P. in which a = 1, d = 1 and n = 50.

$\therefore$ Required Sum = $\frac{n}{2}$ (first term + last term).

= $\frac{50}{2}$ (1 + 50)

= 1275.

Note : Another method to solve the above example :

We know that, $\sum_{i-1}^{n}i$ = $\sum n$ = 1 + 2 + 3 + . . . + n = $\frac{n\left (n+1 \right )}{2}$

Here, 1 + 2 + 3 +…+ 50 = $\frac{50\left (50+1 \right )}{2}$ = 1275

■Remark : If a, b, c are in A.P., we say that b is the arithmetic mean between a and c and, we have : b = $\frac{1}{2}$ (a + c).

■Geometrical Progressions (G.P.) : A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometrical progression.

The constant ratio is called the common ratio of the G.P.

A G.P. with 1^st term = a & common ratio = r is given by :

a, ar, ar², ar³, …

In this G.P., we have : T_n = ar^{n – 1}

Sum of n terms of a G.P. is, S_n = $\frac{a\left (1-r^{n} \right )}{\left (r-1 \right )}$ ; r > 1

= $\frac{a\left (1-r^{n} \right )}{\left (1-r \right )}$ ; r < 1

= n a ; r = 1

= $\frac{r T_{n}-a}{r-1}$ ; r $\ne$ 1.

■ Remark : If a, b, c are in G.P., then b² = ac.

Sum of infinite geometric sequence with the common ratio ( –1 < r < 1 ) = $\frac{a}{1-r}$

■ Harmonical Progression : Numbers a₁, a₂, a₃, ….., a_n are said to be in H.P. if $\frac{1}{a_{1}},\frac{1}{a_{2}},\frac{1}{a_{3}},...\frac{1}{a_{n}}$ are in A.P.

■ Remark : a, b, c are in H.P. if , $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P. and so in this case, b = $\left (\frac{2ac}{a+c} \right )$ .