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.
Find the 25th term of the following arithmetic progression
3, 6, 9, 12, 15, …
3, 6, 9, 12, 15, …
a = 3 , d = 3 , n = 25
Tn = a + (n − 1)d
T25 = 3 + (25 − 1)(3)
= 3 + 72
∴ 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].
Topic : Area of Circle.
Question : A circle with area A1 is contained in the interior of larger circle with area A1+A2. If the radius of the larger circle is 3 and A1,A2,A1+A2 are in A.P,then the radius of smaller circle is__________.
radius of the larger circle = 3
area of the larger circle = A1 + A2
A1+A2 = pi r^2 = 9 pi
A1+A2 = 9 pi —-(1)
A1, A2, A1+A2 are in A.P
hence the difference is common
A2 -A1 = A1 + A2 – A2 = A1
A2 = 2A1 —(2)
solving (1) and ( 2) ,
3A1 = 9 pi -> A1 = 3pi
Area of the smaller circle = 3 pi
radius of the smaller circle = square root of 3
(we have two concentric circles ; A1is the area of the inner circle , A2 is the area between the inner and outer circle)