An arithmetic progression (A.P.) or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13… is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.

Question:-

Find the 25th term of the following arithmetic progression
3, 6, 9, 12, 15, …

Answer:-

3, 6, 9, 12, 15, …

a = 3 , d = 3 , n = 25

Tn = a + (n − 1)d

T25 = 3 + (25 − 1)(3)

= 3 + 72

= 75

∴ the 25th term of the A.P. is 75.

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For determining the length of the chords in Circle, Rectangular Co ordinate System is used. According the graphical representation, radius towards right side is positive and towards left is negative.

Topic : Length of Chords in a Circle of radius r

Using definite integrals we can calculate the length of chords with some constraints.

Question : In a circle of radius r, find the average length of the chords perpendicular to the diameter [-r, r].

Solution :

For more information and clarification related to the above topic you can find help at calculus help.

Topic : Geometric Progression

Problem : The fourth term of a G.P. is 9 and the ninth term is 2,187. Find the first 4 terms of the G.P.

Solution :

Let the G.P. be a, ar, ar² , ar³, …

T₄ = ar³ = 9 ———- (1)

T₈ = ar⁸ = 2,187 —— (2)

(2) ¸(1) :   ar⁸ / ar³ = 2187/9

r⁵ = 243

r⁵ = 3⁵

r = 3

Substitute r = 3 into (1):  a(3³) = 9

a = 1/3

Therefore the first four terms of the G.P. are 1/3, 1/3(3), 1/3(3)², 1/3(3)³,

ie 1/3, 1, 3, 9