Find the sum of first n terms and the sum of first 5 terms of the geometric series 1+32+94+…
Get the complete, step-by-step math solution for: "Find the sum of first n terms and the sum of first 5 terms of the geometric series 1+(2)/(3)+(4)/(9)+". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the first term and common ratio
The given series is a geometric series. The first term, denoted by a, is the first number in the series, which is 1. The common ratio, denoted by r, is found by dividing any term by its preceding term. Here, we divide the second term by the first term: 12/3=32.
Step 2: Formula for the sum of first n terms
The formula for the sum of the first n terms of a geometric series is Sn=1−ra(1−rn), provided that the absolute value of the common ratio r is less than 1. In this case, ∣2/3∣<1, so this formula is applicable.
Step 3: Calculate the sum of first n terms
Substitute the values of a=1 and r=32 into the formula for Sn. Simplify the expression by first calculating the denominator 1−32=31, and then multiplying the numerator by the reciprocal of the denominator.
Step 4: Calculate the sum of first 5 terms
To find the sum of the first 5 terms, substitute n=5 into the derived formula for Sn. Calculate (32)5=3525=24332. Then, perform the subtraction inside the parenthesis and multiply by 3 to get the final sum.