Geometric Progression | Sum of GP Q20

The GMAT Math practice question given below is a sequences and series question. Concept covered is sum of a Geometric Progression and common ratio of a GP and working with algebraic identities. A medium difficulty GMAT 625 level problem solving question.

Question 20: If the ratio of the sum of the first 6 terms of a GP to the sum of the first 3 terms of the GP is 9, what is the common ratio of the Geometric Progression?

1. 3
2. \\frac {1} {3})
3. 2
4. 9
5. \\frac {1} {9})

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Video Explanation

Play Video: GMAT Sum of GP Question

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Explanatory Answer

Step 1 of solving this GMAT Geometric Progressions question:
Formula to find the sum of first 'n' terms of a GP

The sum of the first n terms of a G.P. is given by \\frac {{{ar}^{n}}-a} {r-1}), where 'a' is the first term of the G.P., 'r' is the common ratio and 'n' is the number of terms in the G.P.
Therefore, the sum of the first 6 terms of the G.P will be equal to \\frac {a\left({r}^{6}-1\right)} {r-1})
And sum of the first 3 terms of the G.P. will be equal to \\frac {a\left ( r^3-1\right)} {r-1})

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Step 2 of solving this GMAT Geometric Progressions question:
Use the ratio between these two sums to find 'r'

The ratio of the sum of the first 6 terms : sum of first 3 terms = 9 : 1
i.e., \\frac {\frac {a\left(r^6 - 1\right)} {r - 1}} {\frac {a\left(r^3 - 1\right)} {r - 1}}) = 9

\\frac {{{r}^{6}}-1} {{{r}^{3}}-1}) = \\frac {\left ( {{{r}^{3}}+1} \right )\left ( {{{r}^{3}}-1} \right )} {{{r}^{3}}-1}) = 9

Or r³ + 1 = 9
r³ = 8
r = 2

Choice C is the correct answer.

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Derivation of the Formula to find the Sum of First 'n' terms of a Geometric Progression

Many of us know the formula and may have applied the sum of GP formula many a times. However, we may not have known how the sum of GP formula was derived. Understanding the derivation of the formula for the sum of a finite geometric progression (G.P.) provides valuable insight into how sequences in mathematics operate. Let's break down the formula derivation step by step for clarity and comprehension.

Consider a geometric sequence where:

a is the first term
r is the common ratio, and
n is the number of terms.

The sequence looks like this: (a, ar, ar², ar³, ...... ar^{(n - 1)}

Deriving the Sum of GP Formula

Step 1: Write the Sum Equation

The sum of the first n terms S_n can be expressed as: S_n = a + ar + ar² + ar³ + ..... + ar^{(n - 1)} ---- (1)

Step 2: Multiply both sides of equation (1) by the Common Ratio

rS_n = ar + ar² + ar³ + ...... + arⁿ ---- (2)

Step 3: Perform Subtraction

Subtract the original sum equation (1) from the one obtained by multiplying by r (equation 2):

  rS_n = ar + ar² + ar³ + ...... + arⁿ
- S_n = - (a + ar + ar²+ ..... + ar^{(n - 1)}
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(r - 1)S_n = arⁿ - a
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Or (r - 1)S_n = a(rⁿ - 1)

Step 4: Solve for S_n

Solve the equation for S_n by isolating it on one side:

S_n = \\frac{a \left (r^n - 1 \right )}{r - 1})