Find the sum of the first 15 terms of the geometric series where the first term

Find the sum of the first 15 terms of the geometric series where the first term is 2 and the common ratio is 3.

Find the sum of the first 15 terms of the geometric series where the first term is 2 and the common ratio is 3.

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.

Geometric Series – Understanding the formula for the sum of the first n terms of a geometric series, which involves the first term, common ratio, and number of terms.
Exponents – Calculating powers of the common ratio correctly, especially for larger exponents like 3^15.
Negative Denominator – Recognizing the effect of a negative denominator in the formula and ensuring the signs are handled correctly.