Finite Geometric Series Sum Formula, Derivation and Examples of Use

Deriving the Formula for the Sum of a Geometric Sequence (Part 1)

Consider a geometric sequence with n terms, first term a, and common ratio r:

  a, ar, ar², ..., ar(n-1)
  

We want to find a formula for the sum of this sequence, S:

  S = a + ar + ar² + ... + ar(n-1)
  

Step 1: Multiply the entire sequence by r

  rS = ar + ar² + ar³ + ... + arn
  

Step 2: Align S and rS vertically

  S  =  a + ar    + ar²  + ar³  + ... + ar(n-2) + ar(n-1)
  rS =     ar + ar²  + ar³  + ... + ar(n-2) + ar(n-1) + arn
  

Step 3: Subtract rS from S

Subtracting the sequences, we see that most terms cancel out:

  S - rS = (a + ar + ar² + ... + ar(n-1)) - (ar + ar² + ar³ + ... + arn)
          = a - arn
  

  S - rS = a - arn
  

  S(1 - r) = a - arn
  

  S = (a - arn) / (1 - r)
  

  S = a(1 - rn) / (1 - r)
  

Example: Sum of a Finite Geometric Sequence

Consider a geometric sequence with the first term \( a = 3 \), common ratio \( r = 2 \), and \( n = 5 \) terms:

  3, 6, 12, 24, 48
  

Step 1: Identify the values of a, r, and n

  a = 3, r = 2, n = 5
  

Step 2: Use the formula for the sum of a finite geometric sequence

The formula for the sum of a finite geometric sequence with n terms, first term a, and common ratio r is:

  S = a(1 - rn) / (1 - r)
  

Step 3: Substitute the values into the formula

  S = 3(1 - 25) / (1 - 2)
    = 3(1 - 32) / (-1)
    = 3(-31) / (-1)
    = 93
  

The sum of the first 5 terms of this geometric sequence is 93.

Note: This formula is specifically for finite geometric sequences. For an infinite geometric sequence with \( |r| < 1 \), the sum is given by \( S = a / (1 - r) \).

Example: Sum of a Finite Geometric Sequence with Fractional Common Ratio

Consider a geometric sequence with the first term a = 4, common ratio r = 1/2, and n = 6 terms:

  4, 2, 1, 1/2, 1/4, 1/8
  

Step 1: Identify the values of a, r, and n

  a = 4, r = 1/2, n = 6
  

Step 2: Use the formula for the sum of a finite geometric sequence

  S = a(1 - rⁿ) / (1 - r)
  

Step 3: Substitute the values into the formula

  S = 4(1 - (1/2)⁶) / (1 - 1/2)  # Substitute a, r, and n
    = 4(1 - (1/64)) / (1/2)      # Calculate (1/2)⁶
    = 4(63/64) / (1/2)           # Subtract 1/64 from 1
    = (4 * 63/64) / (1/2)        # Multiply 4 by 63/64
    = (4 * 63/64) * 2            # Divide by 1/2 is the same as multiplying by 2
    = 4 * 63/32                  # Simplify the fraction
    = 252/32                     # Multiply 4 by 63
    = 63/8                       # Divide by 4
    = 7.875                      # Divide 63 by 8
  

The sum of the first 6 terms of this geometric sequence is 63/8 or 7.875.

Example: Sum of a Finite Geometric Sequence with Decimal Common Ratio

Consider a geometric sequence where we want to explore the effect of a small common ratio, such as r = 0.1. This choice of r means that each term is 10% of the previous term, leading to a rapid decrease in the value of the terms. Let’s take the first term a = 5, common ratio r = 0.1, and n = 4 terms:

  5, 0.5, 0.05, 0.005  # Each term is 10% of the previous term
  

Step 1: Identify the values of a, r, and n

  a = 5, r = 0.1, n = 4  # First term is 5, common ratio is 0.1, and there are 4 terms
  

Step 2: Use the formula for the sum of a finite geometric sequence

  S = a(1 - rⁿ) / (1 - r)  # General formula for the sum of a finite geometric sequence
  

Step 3: Substitute the values into the formula

  S = 5(1 - 0.1⁴) / (1 - 0.1)  # Substitute a, r, and n
    = 5(1 - 0.0001) / 0.9      # Calculate 0.1⁴, which represents the ratio raised to the power of 4
    = 5(0.9999) / 0.9          # Subtract 0.0001 from 1
    = 4.9995 / 0.9             # Multiply 5 by 0.9999
    = 5.555                   # Divide 4.9995 by 0.9
  

The sum of the first 4 terms of this geometric sequence is 5.555. The choice of r = 0.1 leads to a sequence where each term is only 10% of the previous term, resulting in a quickly diminishing sequence.

Finding the Number of Terms \( n \) in a Geometric Sequence

Given the sum \( S \), the first term \( a \), and the common ratio \( r \), you can find the number of terms \( n \) using the following steps:

1. Start with the sum formula: \[ S = \frac{{a(1 – r^n)}}{{1 – r}} \]
2. Multiply both sides by \( 1 – r \): \[ S(1 – r) = a(1 – r^n) \]
3. Divide both sides by \( a \): \[ \frac{{S(1 – r)}}{a} = 1 – r^n \]
4. Rearrange to isolate \( r^n \): \[ r^n = 1 – \frac{{S(1 – r)}}{a} \]
5. Take the natural logarithm of both sides: \[ \ln(r^n) = \ln\left(1 – \frac{{S(1 – r)}}{a}\right) \]
6. Use the property of logarithms that \(\ln(x^y) = y \cdot \ln(x)\): \[ n \cdot \ln(r) = \ln\left(1 – \frac{{S(1 – r)}}{a}\right) \]
7. Solve for \( n \): \[ n = \frac{{\ln\left(1 – \frac{{S(1 – r)}}{a}\right)}}{{\ln(r)}} \]

Important Consideration

Keep in mind that the value of \( n \) should be a positive integer, as it represents the number of terms in the sequence. If you obtain a non-integer value for \( n \), such as 2.4, it may indicate one of the following:

• Inconsistency: The given information (sum, first term, common ratio) may be inconsistent, leading to an invalid result for \( n \).
• Problem Definition: The problem may not be well-defined, and additional information or constraints may be required to obtain a valid solution for \( n \).

Always verify the given information and ensure that the problem is properly defined before attempting to solve for \( n \) in a geometric sequence.