To find the sum of the first \( n \) terms of a geometric sequence, we use one of the provided formulas. The correct formula among the options given is:
[tex]\[ S_n = a_1 \left( \frac{1 - r^n}{1 - r} \right) \][/tex]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the number of terms.
Let’s explain how to use this formula to find the sum of the pennies on Rows 1-4 and understand the importance of using the formula:
Step-by-Step Calculation Using the Formula:
1. Identify the first term \( a_1 \):
- This is the first number in the sequence. Let's assume the geometric sequence starts with 1 penny on the first row.
- Hence, \( a_1 = 1 \).
2. Determine the common ratio \( r \):
- The common ratio is the factor by which each term is multiplied to get the next term in the sequence. Suppose each row doubles the number of pennies from the previous row (i.e., \( r = 2 \)).
3. Count the number of terms \( n \):
- For Rows 1-4, we have \( n = 4 \).
4. Insert the values into the formula:
- Substitute \( a_1 = 1 \), \( r = 2 \), and \( n = 4 \) into the formula:
[tex]\[
S_4 = 1 \left( \frac{1 - 2^4}{1 - 2} \right)
\][/tex]
5. Simplify the formula:
- Calculate the powers and the fractions:
[tex]\[
S_4 = 1 \left( \frac{1 - 16}{1 - 2} \right)
\][/tex]
- Simplify inside the parentheses:
[tex]\[
S_4 = 1 \left( \frac{-15}{-1} \right)
\][/tex]
- Simplify the fraction:
[tex]\[
S_4 = 1 \times 15
\][/tex]
- Calculate the result:
[tex]\[
S_4 = 15
\][/tex]
Why Use the Formula?
1. Efficiency:
- The formula allows you to quickly calculate the sum of a series without having to manually add each term.
2. Accuracy:
- Using the formula reduces the risk of making arithmetic errors, especially with long sequences or large numbers.
3. Generality:
- The formula is applicable to any geometric sequence, making it a versatile tool in various mathematical contexts.
Thus, using the formula [tex]\( S_n = a_1 \left(\frac{1 - r^n}{1 - r}\right) \)[/tex] efficiently computes the sum of the first [tex]\( n \)[/tex] terms of a geometric sequence.
