Finding the Sum of the First 7 Terms of a Geometric Progression

Geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this article, we will explore how to find the sum of the first 7 terms of a GP where the first term is 729 and the 7th term is 64.

Understanding the Problem

We are given the first term a 729 and the 7th term a_7 64. The goal is to find the common ratio ( r ) and then use it to determine the sum of the first 7 terms, denoted as ( S_7 ).

Step 1: Finding the Common Ratio (r)

The general term of a GP is given by the formula:

a_n a cdot r^{(n-1)}

Using this formula for the 7th term, we have:

a_7 a cdot r^6 64

Substituting ( a 729 ) into the equation:

729 cdot r^6 64

To find ( r^6 ), we divide both sides by 729:

r^6 frac{64}{729}

We simplify the right-hand side:

r^6 left(frac{4}{9}right)^3

Therefore:

r left(frac{4}{9}right)^{frac{1}{6}} frac{2}{3}

Step 2: Calculating the Sum of the First 7 Terms

The sum of the first ( n ) terms of a GP is given by the formula:

S_n a cdot frac{1 - r^n}{1 - r} for ( r eq 1 )

For ( n 7 ), the formula becomes:

S_7 729 cdot frac{1 - left(frac{2}{3}right)^7}{1 - frac{2}{3}}

We simplify this step by step:

S_7 729 cdot frac{1 - left(frac{2}{3}right)^7}{frac{1}{3}}

Next, we calculate (left(frac{2}{3}right)^7):

left(frac{2}{3}right)^7 frac{128}{2187}

Substituting this into the formula:

S_7 729 cdot frac{1 - frac{128}{2187}}{frac{1}{3}}

Simplifying the numerator:

frac{2187 - 128}{2187} frac{2059}{2187}

Therefore, the formula becomes:

S_7 729 cdot 3 cdot frac{2059}{2187}

Further simplifying:

S_7 2187 cdot frac{2059}{2187} 2059

Hence, the sum of the first 7 terms of the GP is 2059.

Verification

To verify the result, we can calculate the individual terms and sum them up:

1st term: 729 2nd term: 729 cdot frac{2}{3} 486 3rd term: 486 cdot frac{2}{3} 324 4th term: 324 cdot frac{2}{3} 216 5th term: 216 cdot frac{2}{3} 144 6th term: 144 cdot frac{2}{3} 96 7th term: 96 cdot frac{2}{3} 64

Summing these terms:

729 486 324 216 144 96 64 2059

This confirms that the sum of the first 7 terms of the GP is indeed 2059.