Question

Arithmetic progression geometric progression and harmonic progression​

Answer 1

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.

Eg   2,4,6,8,10

the common difference here is 2

an geometric progression is a sequence of number  where each term after the first is found by multiplying the previous one by the common ratio

Eg 4,12,36....

the common ratio is 3

harmonic progression​ =1/Arithmetic progression

Answer 2

Answer:

Geometric Progression

A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. For example, the sequence 4, -2, 1, - 1/2,.... is a Geometric Progression (GP) for which - 1/2 is the common ratio.

The general form of a GP is a, ar, ar2, ar3 and so on.

The nth term of a GP series is Tn = arn-1, where a = first term and r = common ratio = Tn/Tn-1) .

The formula applied to calculate sum of first n terms of a GP:

When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac

The sum of infinite terms of a GP series S∞= a/(1-r) where 0< r<1.

If a is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = arm-n.

The nth term from the end of the G.P. with the last term l and common ratio r is l/(r(n-1)) .

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term is derived from the preceding term by adding or subtracting a fixed number called the common difference "d"

For example, the sequence 9, 6, 3, 0,-3, .... is an arithmetic progression with -3 as the common difference. The progression -3, 0, 3, 6, 9 is an Arithmetic Progression (AP) with 3 as the common difference.

The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on. Thus nth term of an AP series is Tn = a + (n - 1) d, where Tn = nth term and a = first term. Here d = common difference = Tn - Tn-1.

Sum of first n terms of an AP: S =(n/2)[2a + (n- 1)d]

The sum of n terms is equal to the formulawhere l is the last term.

Tn = Sn - Sn-1 , where Tn = nth term

When three quantities are in AP, the middle one is called as the arithmetic mean of the other two. If a, b and c are three terms in AP then b = (a+c)/2

Harmonic Progression

A series of terms is known as a HP series when their reciprocals are in arithmetic progression. Example: 1/a, 1/(a+d), 1/(a+2d), and so on are in HP because a, a + d, a + 2d are in AP. The nth term of a HP series is Tn =1/ [a + (n -1) d].

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