Sum of GP
A Sum of GP calculator computes the sum of a given number of terms in a geometric progression. It uses the formula \( S_n = a \frac{1 – r^n}{1 – r} \) for finite series and \( S_{\infty} = \frac{a}{1 – r} \) for infinite series, ensuring accurate and quick results by inputting the first term, common ratio, and number of terms.
Classroom Note: Sum of a Geometric Progression (GP)
Introduction to Geometric Progression (GP)
A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This ratio remains constant throughout the sequence.
Definition
If the first term of a GP is \( a \) and the common ratio is \( r \), then the GP can be written as:
\[ a, ar, ar^2, ar^3, \ldots \]
Sum of the First \( n \) Terms of a GP
The sum of the first \( n \) terms of a GP is denoted by \( S_n \). There are different formulas to find \( S_n \), depending on the value of the common ratio \( r \).
When \( r \neq 1 \)
If \( r \neq 1 \), the sum of the first \( n \) terms of a GP is given by:
\[ S_n = a \frac{1-r^n}{1-r} \]
Derivation of the Formula:
1. Write the sum of the first \( n \) terms:
\[ S_n = a + ar + ar^2 + ar^3 + \cdots + ar^{n-1} \]
2. Multiply both sides of the equation by \( r \):
\[ rS_n = ar + ar^2 + ar^3 + \cdots + ar^{n-1} + ar^n \]
3. Subtract the second equation from the first:
\[ S_n – rS_n = a – ar^n \]
\[ S_n(1 – r) = a(1 – r^n) \]
4. Solve for \( S_n \):
\[ S_n = \frac{a(1 – r^n)}{1 – r} \]
When \( r = 1 \)
If \( r = 1 \), each term of the GP is the same as the first term \( a \). Therefore, the sum of the first \( n \) terms is:
\[ S_n = a + a + a + \cdots + a = na \]
Sum of an Infinite GP
If \( |r| < 1 \), the sum of an infinite GP can be found using the formula:
\[ S_{\infty} = \frac{a}{1 – r} \]
Vital Tips for Better Understanding:
1. Identify the First Term and Common Ratio:
– Always start by identifying the first term \( a \) and the common ratio \( r \) of the GP.
2. Check the Value of \( r \):
– Use the formula \( S_n = \frac{a(1 – r^n)}{1 – r} \) for \( r \neq 1 \).
– Use \( S_n = na \) for \( r = 1 \).
– For infinite series, ensure \( |r| < 1 \) to apply \( S_{\infty} = \frac{a}{1 – r} \).
3. Simplify Expressions:
– Practice simplifying expressions involving powers and fractions, as these skills are often needed in calculations involving GPs.
Example
Let’s find the sum of the first 5 terms of a GP where \( a = 3 \) and \( r = 2 \).
1. Identify the terms:
– The first 5 terms are: \( 3, 6, 12, 24, 48 \).
2. Use the formula:
\[ S_5 = 3 \frac{1 – 2^5}{1 – 2} \]
3. Simplify:
\[ S_5 = 3 \frac{1 – 32}{1 – 2} = 3 \frac{-31}{-1} = 3 \times 31 = 93 \]
So, the sum of the first 5 terms is \( 93 \).
Example of Infinite GP
Find the sum of the infinite GP where \( a = 5 \) and \( r = \frac{1}{3} \).
1. Ensure \( |r| < 1 \):
– Since \( \left|\frac{1}{3}\right| < 1 \), we can use the infinite sum formula.
2. Use the formula:
\[ S_{\infty} = \frac{5}{1 – \frac{1}{3}} \]
3. Simplify:
\[ S_{\infty} = \frac{5}{\frac{2}{3}} = 5 \times \frac{3}{2} = \frac{15}{2} = 7.5 \]
So, the sum of the infinite GP is \( 7.5 \).
Recap
– Geometric Progression (GP): Sequence with a constant ratio between terms.
– Sum of First \( n \) Terms:
– \( S_n = \frac{a(1 – r^n)}{1 – r} \) for \( r \neq 1 \).
– \( S_n = na \) for \( r = 1 \).
– Sum of Infinite GP: \( S_{\infty} = \frac{a}{1 – r} \) for \( |r| < 1 \).
Understanding and applying these formulas will help in solving problems related to geometric progressions effectively.
