what is math formula for 1+1=2,2+2=4...up to 240

GP Sum

The sum of a GP is the sum of a few or all terms of a geometric progression. A series of numbers obtained by multiplying or dividing each preceding term, such that there is a common ratio between the terms (that is not equal to 0) is the geometric progression and the sum of all these terms formed so is the sum of geometric progression (GP).

Allow us learn more most GP sum formulas (for both finite and infinite series) along with examples.

1.	What is GP Sum?
2.	Sum of n Terms in GP
3.	Sum of Infinite GP
4.	Sum of GP Formulas
5.	FAQs on GP Sum

What is GP Sum?

GP sum is the sum of a few or all terms of a geometric progression. Let u.s. start understanding GP sum using an instance. Clara saves a few dollars every week in a detail fashion. In week 1 she deposits $2. In calendar week 2 - $iv, in week 3 - $eight, in week 4 - $16, and and then on. How much will she have at the cease of 6 weeks in her piggy bank? The corporeality deposited in the six weeks would be $two, $4, $viii, $16, $32, and $64. So, at the end of half dozen weeks she will have $two+ $iv + $8+ $16+ $32 + $64 = $126 accumulated in her piggy bank. Since we had to summate the amount simply later 6 weeks, the manual adding was like shooting fish in a barrel. But what if we had to notice the amount later on so many weeks, say 50 weeks? GP sum formulas make this process easier.

Sum of n Terms in GP

Let's discuss how to calculate the sum of n terms of GP. Consider the sum of the first northward terms of a geometric progression (GP) with first term a and common ratio r. Then the start 'n' terms of GP are of the form a, ar, ar², ... ar^{due north-one}. Permit S exist the sum of the geometric progression of n terms. And then:

S_{due north} = a + ar + ar^two + ... + ar^n-i ... (ane)

Multiply both sides past r:

rS_n = ar + ar² + ... + ar^northward ... (2)

Subtracting equation (1) from equation (2):

rS_northward - Due south_n = (ar + ar^two + ... + ar^{due north}) - (a + ar + arⁱⁱ + ... + ar^{due north-i})

Due south_n (r - 1) = ar^northward - a

S_n (r - 1) = a(rⁿ - 1)

S_north = a(r^{due north} - i) / (r - 1)

Note that, here, r ≠ 1. This formula can also exist written as:

Southward_{due north} = -a(ane - r^northward) / (-(1 - r)) = a(1 - rⁿ) / (1 - r).

What happens when r = 1? Then the GP is of form a, a, a, ..., (north terms). Then the sum is, S_north = na.

Now, let usa work on the example (from the final section) using the sum of due north terms of GP formula. The amounts saved by Clara in the club of weeks are, 2, 4, 8, sixteen, .... Clearly, this is a geometric progression as iv/2 = viii/four = 16/2 = ... = 2.

Here, the first term is, a = 2, the common ratio is, r = 2, and the number of terms is, n = half-dozen (as we want the sum of the amounts after half-dozen weeks).

We have got the same answer using the GP sum formula also.

Sum of Space GP

The higher up formula gives the sum of a finite GP. But what if nosotros have to find the sum of an infinite GP? Let united states of america consider a GP a, ar, ar², ... (up to an infinite number of terms) such that the accented value of its common ratio is less than 1. i.e., |r| < one. Let the states assume the sum of all these infinite number of terms be Southward_∞. And so:

S_∞ = a + ar + arⁱⁱ + ... ... (1)

Multiplying both sides by r,

rS_∞ = ar + ar² + ... ... (2)

Subtracting (two) from (1):

(i - r) S_∞ = a

S_∞ = a / (one - r)

This is the formula for the sum of infinite gp. Note that for any infinite geometric serial,

• If |r| < 1, then the geometric series converges and information technology has a sum.
• If |r| ≥ 1, then the geometric series diverges and information technology cannot have a sum.

Example: A square is drawn by joining the midpoints of the sides of the original square. A third foursquare is drawn inside the 2nd foursquare in the aforementioned manner and this process is continued indefinitely. If a side of the first square is "s" units, decide the sum of areas of all the squares then formed?

Solution:

The areas of squares thus formed are, s, south²/ii, s²/four, s²/8, ....

Taking 's²' every bit common gene, the sum of areas is, sⁱⁱ ( i + 1/two + ane/4 + ... ) ... (1)

Hither, 1 + 1/2 + one/4 + ... is an space geometric serial with a = 1 and r = 1/2. And so its sum is

Southward_∞ = a / (1-r) = 1 / (1 - 1/ii) = 2.

Substituting this in (ane):

The sum of areas = 2s².

Sum of GP Formulas

Let us summarize all formulas used for finding the sum of a GP.

• To find the sum of finite (n) terms of a GP,
  S_{due north} = a(r^north - one) / (r - i) [OR] S_n = a(1 - r^north) / (1 - r), if r ≠ i.
  Southward_northward = an, if r = i.
• To find the sum of space terms of a GP,
  S = a / (1 - r), if |r| < 1 (and in this case, we say that the series converges)
  Southward cannot be plant if |r| ≥ 1 (and in this example, we say that the series diverges)

These GP sum formulas are summarized in the flowchart below.

Important Notes on GP Sum:

• The sum of GP (of north terms) is: South_n = a(rⁿ - 1) / (r - 1) [OR] Due south_north = a(1 - rⁿ) / (ane - r), if r ≠ 1.
• The sum of GP (of northward terms) is: Due south_n = na, when r = 1.
• The sum of GP (of infinite terms) is: Due south_∞ = a/(1-r), when |r| < ane.
• The sum of GP (of infinite terms) is: S_∞ = does non exist, when |r| ≥ one.

☛ Related Topics:

• n^thursday Term of a GP
• Geometric Sequence Calculator
• Infinite Geometric Series Figurer
• Geometric Sequence Formulas

GP Sum Examples

1. Instance 1: In a GP, the sum of the commencement three terms is 16, and the sum of the next 3 terms is 128. Find the sum of the first north terms of the GP.

   Solution:

   Let 'a' and 'r' be the first term and the common ratio of the given GP respectively. Then:

   a + ar + ar² = 16
   ar^three + ar⁴ + ar⁵ = 128

   Nosotros tin rewrite these equations every bit:

   a (i + r + r²) = 16
   ar^three (1 + r + r²) = 128

   Dividing the second equation past offset,

   [ar^three (1 + r + r²)] / [a (1 + r + r²)] = 128/16

   r³ = 8

   r = 2

   Substituting this in the first equation:

   a (one + two + 2²) = sixteen

   7a = 16

   a = sixteen/7

   The sum of n terms in GP is:

   S_n = a(rⁿ - 1) / (r - 1) = [ (16/7) (two^northward - 1) ] / (2 - 1) = 16 (twoⁿ - one) / 7

   Answer: Southward_n = 16 (2^northward - 1) / 7.

2. Case ii: Sara shared a message with five unique people at 1 am. At 2 am, each of her friends shared it with five unique people. So at 3 pm each of their friends shared with 5 unique people. In this sequence, how many unique people would have received the message by eight am?

   Solution:

   The number of unique people who received letters afterward each 60 minutes starting at 1 AM are 5, 25, 125, ...

   Conspicuously, this is a GP with a = 5 and r = v. We take to discover the GP sum for eight terms (8 PM).

   Using the sum of GP formula,

   Southward_north = a(r^{due north} - 1) / (r - 1)

   Southward₈ = 5(5⁸ - ane) / (5 - 1) = 488,280

   Reply: The total number of unique people = 488,280.

3. Example 3: The altitude traveled by a ball dropped from a pinnacle (in inches) are 128/ix, 32/3, 8, half dozen... What could be the distance traveled by the brawl before coming to remainder?

   Solution:

   The distances traveled by the brawl = 128/9, 32/3, viii, 6....

   This is a GP considering (32/three) / (128/9) = 8 / (32/3) = six/8 = ... = three/4.

   So the first term is, a = 128/9; and the mutual ratio is, r = iii/4.

   The full distance traveled by the ball will exist the sum of this infinite GP.

   S_∞ = a/(1-r)

   = (128/9) / (1 - (3/4))

   = (128/9) / (one/iv)

   = 128/9 × iv/1

   = 56.89 inches

   Answer: The total distance traveled by the ball = 56.89 inches.

become to slidego to slidego to slide

Ready to see the world through math'southward optics?

Math is a life skill. Assist your child perfect it through real-world awarding.

Volume a Complimentary Trial Class

Exercise Questions on Sum of GP Formula

go to slidego to slide

FAQs on GP Sum

What is the Formula of Finding GP Sum for Finite Terms?

The GP sum formula used to find the sum of n terms in GP is, Southward_n = a(r^{due north} - i) / (r - 1), r ≠ 1 where:

• a = the first term of GP
• r = the common ratio of GP

If r = 1, then S_n = na.

When Does the Sum of an Space GP Diverge?

The sum of an infinite GP diverges when the accented value of the common ratio of GP is either equal to 1 or greater than 1. i.eastward., when |r| ≥ 1.

How to Discover the Sum of n Terms in GP?

The sum of n terms in GP with 'a' to exist its first term and 'r' to be its mutual ratio can be plant using one of the formulas:

• S_n = a(rⁿ - 1) / (r - ane)
• South_n = a(1 - rⁿ) / (1 - r)

These 2 formulas would work when r ≠ one. If r = i, then the sum of n terms is S_n = na.

What is the Sum of Geometric Progression Formula for Infinite Terms?

If GP is infinite, then we can find its sum but when |r| < 1 and nosotros use the formula S_∞ = a/(1-r) in this case. In case, |r| ≥ 1 for an space GP, then it diverges and hence nosotros cannot find its sum.

What is the Sum of GP with r = ane?

If r = 1, so the GP is of form a, a, a, ... The sum of finite (n) terms of such GP = a + a + a + ... (n times) = na. Nosotros cannot find the sum of space terms of such GP.

When Does the Sum of an Infinite GP Converge?

The sum of an infinite GP converges when the absolute value of the common ratio of GP is less than i. i.e., when |r| < ane.

scherereccoved.blogspot.com

Source: https://www.cuemath.com/algebra/sum-of-a-gp/

Share this post