CAT To find the sum of n terms of a GP MCQ - Practice Questions & Answers

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3 Questions around this concept.

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Find $4+12+36+\ldots$ up to 6 terms.

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Concepts Covered - 1

To find the sum of n terms of a GP

Formula to find the sum of the first n terms of a GP (when $\mathrm{r\neq 1}$ ):

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

Where:

- $\mathrm{S_n}$ is the sum of the first n terms

- a is the first term

- r is the common ratio

- n is the number of terms

If r = 1, then $\mathrm{S_n=n\times a}$

Foundation Building Questions:

Question 1: Given a GP whose first term is 2 and the common ratio is 3, find the sum of the first 4 terms.

Solution:

Using the formula:

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

For a=2, r=3, and n=4:

$\mathrm{ S_4=\frac{2\left(1-3^4\right)}{1-3} }$

$\mathrm{ S_4= \frac{2(-80)}{-2} }$

$\mathrm{ S_4= 80 }$

Question 2: The sum of the first three terms of a GP is 14, and the sum of the next three terms is 98. Determine the GP.

Solution:

Let's assume the first term is a and the common ratio is r.

For first three terms:

$\mathrm{ S_3=\frac{a\left(1-r^3\right)}{1-r}=14 }$

For next three terms (from 4th to 6th term):

Using our formula and understanding from previous concepts, the sum of terms from 4th to 6th would be:

$\mathrm{ S_3^{\prime}=\frac{a \times r^3\left(1-r^3\right)}{1-r}=98}$

Using these equations, we can derive the values of a and r and thus, find the GP.

Application of Previous Concepts:

To determine the sum of terms in a GP, it's imperative first to establish that the sequence given is a GP. This could be done by ensuring the ratio between any two consecutive terms remains consistent (from Concept 1). Additionally, once you determine individual terms (using Concept 4), the application of the sum formula becomes straightforward.

Tips and Tricks:

1. Utilise the Nature of GP: The terms in a GP can grow very rapidly (for r > 1) or diminish towards zero (for 0 < r < 1). Recognizing the nature of the GP can help in estimating the sum.

2. Factorization Trick: Sometimes, GP sum questions can be solved more easily by factorizing terms or using properties of exponents.

3. Integration with AP: Some challenging problems might integrate both GP and AP concepts. For instance, a sequence might have terms derived from both progressions. Be prepared to identify and solve such integrated problems.

4. Series Splitting: For problems where parts of the series are given (like in Question 2 above), split the series accordingly to derive the needed parameters (like a or r).

5. Real-world Application: Remember, GPs frequently appear in real-world scenarios like compound interest, finance problems involving repeated multiplication factors, biology in population growth, physics in decay processes, etc. Identifying GP in such contexts can be half the battle.

6. Practice: As with all quantitative topics, there's no substitute for practice. The more problems you solve related to GP sums, the more patterns, shortcuts, and techniques you'll discover.

Always approach GP sum problems systematically. Understand the problem's requirements, ensure you've correctly identified the GP, and then apply the appropriate formula or strategy. Being methodical reduces the chances of errors and increases speed.