## 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.