# Geometric Progression | Sum of GP Q20

#### GMAT Sample Questions | GMAT Quant Question Bank

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. $$frac {1} {3}$ 3. 2 4. 9 5. $\frac {1} {9}$ ## Get to 705+ in the GMAT #### Online GMAT Course @ INR 8000 + GST ### Video Explanation Play Video: GMAT Sum of GP Question ## GMAT Live Online Classes #### Starts Sat, June 29, 2024 ### Explanatory Answer #### Step 1 of solving this GMAT Geometric Progressions question: Formula to find the sum of first 'n' terms of a GP The sum of the first n terms of a G.P. is given by $\frac {{{ar}^{n}}-a} {r-1}$, where 'a' is the first term of the G.P., 'r' is the common ratio and 'n' is the number of terms in the G.P. Therefore, the sum of the first 6 terms of the G.P will be equal to $\frac {a\left${r}^{6}-1$right$} {r-1}) And sum of the first 3 terms of the G.P. will be equal to $\frac {a\left$ r^3-1$right$} {r-1}) #### Step 2 of solving this GMAT Geometric Progressions question: Use the ratio between these two sums to find 'r' The ratio of the sum of the first 6 terms : sum of first 3 terms = 9 : 1 i.e., $\frac {\frac {a\left$r^6 - 1$right$} {r - 1}} {$frac {a\left$r^3 - 1$right$} {r - 1}}) = 9 $\frac {{{r}^{6}}-1} {{{r}^{3}}-1}$ = $\frac {\left$ {{{r}^{3}}+1} $right$$left$ {{{r}^{3}}-1} $right$} {{{r}^{3}}-1}) = 9 Or r3 + 1 = 9 r3 = 8 r = 2 #### Choice C is the correct answer. ### 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 Sn can be expressed as: Sn = a + ar + ar2 + ar3 + ..... + ar(n - 1) ---- (1)

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

rSn = 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):

rSn = ar + ar2 + ar3 + ...... + arn
- Sn = - (a + ar + ar2+ ..... + ar(n - 1)
------------------------------------------------------------------
(r - 1)Sn = arn - a
------------------------------------------------------------------

Or (r - 1)Sn = a(rn - 1)

Step 4: Solve for Sn

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

Sn = $$frac{a \left$r^n - 1 $right$}{r - 1}) #### GMAT Online CourseTry it free! Register in 2 easy steps and Start learning in 5 minutes! #### Already have an Account? #### GMAT Live Online Classes Next Batch June 29, 2024 Work @ Wizako ##### How to reach Wizako? Mobile:$91) 95000 48484
WhatsApp: WhatsApp Now
Email: learn@wizako.com