Geometric Series Sum Formula

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Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.

A sequence a1, a2, a3, …, an, … is called geometric progression, if each term is non-zero and

211

By letting a1 = a, we obtain a geometric progression, a, ar, ar2, ar3,…., where a is called the first term and r is called the common ratio of the G.P.

Example: Find whether the series 2,4,8,16,....... is in G.P ?

here a1 = 2, a2 = 4, a3 = 8, a4 = 16

a2/ a1 =4/2 = 2 ; a3/a2 = 8/4 =2 ; a4/a3 =16/8 = 2

so the common ratio r = 2.

hence the series is in Geometric progression.

The nth term of geometric progression :

Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. Write a few terms of it. The second term is obtained by multiplying a by r, thus a2 = ar. Similarly, third term is obtained by multiplying a2 by r. Thus ...
... a3 = a2r = ar2, and so on.

1st term = a1 = a = ar1-1 ; 2nd term = a2 = ar = ar2-1 ; 3rd term = a3 = ar2 = ar3-1 ;
4th term = a4 = ar3 = ar4-1 ; 5th term = a5 = ar4 = ar5-1 ;

Therefore, the pattern suggests that the nth term of a G.P. is given by

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Finite Geometric series :

Geometric Progression can be written as a, ar, ar2, ar3, … arn-1 ; a, ar, ar2,...,arn – 1.......;according as G.P. is finite or infinite, respectively.
The series a + ar + ar2 + ... + arn–1 or a + ar + ar2 + ... + arn–1 +..........are called finite or infinite geometric series, respectively.

So for finite series a k+1 / a k = a k / a k-1 for every natural k < n-1

Sum of Finite Geometric Series :

Let the first term of a G.P. be a and the common ratio be r. Let us denote by Sn the sum to first n terms of G.P. Then

Sn = a + ar + ar2 +...+ arn–1 ................. (1)
Case 1 : If r = 1, we have Sn = a + a + a + ... + a (n terms) = na
Case 2 : If r ≠ 1, multiplying (1) by r, we have
rSn = ar + ar2 + ar3 + ... + arn ..............(2)

Subtracting (2) from (1),

we get (1 – r) Sn = a – arn = a(1 – rn)

This gives us Sum of n terms Sn

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Example

Example 1 Find the sum of first n terms and the sum of first 5 terms of the geometricseries 1 + 2/3 + 4/9 +...........

Here a = 1 and r = 2/3 . Therefore

finite series

Example 2:

Find the 12th term of the series:

3,6,12,24,48…

Solution:

The given series is a geometric progression with a=3,r=(6)/(3) =2

T12=ar11=3 x 211=6144

The midpoint is the middle point of a line segment. It is equidistant from both endpoints it was introduced by the scientist duct, the midpoint consists of a many points of a triangle as well as a quadrilateral, and midpoint follows the specific line segment in a triangle. (Source: Wikipedia)

Definition: A technique used to make the coefficient level and the level may be average elasticity for discrete changes in two variables, C and D. The described characteristic of this formula is that percentages changes are calculate based on average of the ending and initial values of each variable, rather than initial values.

Midpoint Formula: If (x1,y1) and (x2,y2) are the two end points of the line segment,then the midpoint of the line segment is:

{ (x1+x2)/2 , (y1+y2)/2 }

Midpoint Explained

Use of midpoint formula: The midpoint formula is used to find the accurate Value between the two points; it follows the tangential line segment and the interior region of the minor axis.

Midpoint astrology: The midpoint is nothing but a maths point halfway between two specified level bodies that is use to tell about an individual of the inter modulated picture. The consider point will differ in this units.

Midpoint types: There are two types in it; they are indirect midpoint and direct midpoint. The indirect midpoint occur when the unleveled temperature is detected through various situations in the consideration bodies .the direct level may be completely differ by the products present in the static in it.

Midpoint Examples

Ex 1: Find the midpoint value for the given (-2, 4) and (6,-10)

Sol: Where the x1 and x2 divided by 2 values and add with y1 and y2 divided by 2 value (x1+x2)/2 ,(y1+y2)/2 .the x1 and x2 are the two quadrants level points in the system mentioned as :
(-2+6)/2,(4-10)/2

{2,-3}

So the answer for the above sum is (2,3).

Ex 2 : Find the value of p so that (-4, 4.5) is the mid point of (p, 4) and (-3, 6).

Sol: By using mid point (x1+x2)/2 ,(y1+y2)/2 This reduces to needing to figure out what r is, in order to make the x values work:

(p-3)/2 ,(4+6)/2

(P-3)/2 = -4

P-3 = -8

P = -5 The answer is -5.

For more you can connect to an online tutor anytime and get the required help. There is also an online midpoint formula calculator provided in this page for a better understanding.

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