Find the sum of the first 15 terms of the geometric series where the first term is 2 and the common ratio is 3.
Practice Questions
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Q1
Find the sum of the first 15 terms of the geometric series where the first term is 2 and the common ratio is 3.
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The sum of the first n terms of a geometric series is S_n = a(1 - r^n) / (1 - r). Here, a = 2, r = 3, n = 15. So, S_15 = 2(1 - 3^15) / (1 - 3) = 2(1 - 14348907) / -2 = 14348906.
Questions & Step-by-step Solutions
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Q
Q: Find the sum of the first 15 terms of the geometric series where the first term is 2 and the common ratio is 3.
Solution: The sum of the first n terms of a geometric series is S_n = a(1 - r^n) / (1 - r). Here, a = 2, r = 3, n = 15. So, S_15 = 2(1 - 3^15) / (1 - 3) = 2(1 - 14348907) / -2 = 14348906.
Steps: 9
Step 1: Identify the first term (a) of the geometric series. Here, a = 2.
Step 2: Identify the common ratio (r) of the geometric series. Here, r = 3.
Step 3: Identify the number of terms (n) you want to sum. Here, n = 15.
Step 4: Use the formula for the sum of the first n terms of a geometric series: S_n = a(1 - r^n) / (1 - r).
Step 5: Substitute the values into the formula: S_15 = 2(1 - 3^15) / (1 - 3).
Step 6: Calculate 3^15. This equals 14348907.
Step 7: Substitute 3^15 back into the equation: S_15 = 2(1 - 14348907) / (1 - 3).
Step 8: Simplify the equation: S_15 = 2(-14348906) / -2.
Step 9: The -2 in the numerator and denominator cancel out, leaving S_15 = 14348906.
