CAT To find the nth term of a GP MCQ - Practice Questions & Answers

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3 Questions around this concept.

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Find the 5^th term in the series 5, 15, 45, _____

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Concepts Covered - 1

To find the nth term of a GP

Formula to find the nth term of a GP:

$\mathrm{a_n=\mathrm{a} \times r^{(n-1)}}$

Where:

- $\mathrm{a_n}$ is the nth term

- a is the first term

- r is the common ratio

- n is the number of terms

Foundation Building Questions:

Question 1: The second term of a GP is 6 and the fourth term is 54. Determine the GP.

Solution:

Using the formula $\mathrm{a_n = a \times r^{(n-1)}:}$

For $\mathrm{ n=2, a_2 = a \times r = 6}$

For $\mathrm{n=4,a_4 = a \times r^{3} = 54}$

From the first equation, we get $\mathrm{a = 6/r}$

Plugging this into the second equation and simplifying, we derive r = 3. Thus, a = 2.

The GP is 2, 6, 18, 54,...

Question 2: If the nth term of a GP is 16 and the (n+2)th term is 64, find the common ratio, 'r'.

Solution:

Using the formula $\mathrm{{ a_n }=a \times r^{\wedge}\{(n-1)\}}$

For $\mathrm{a_n, \text { ax } r^{(n-1)}=16}$

For $\mathrm{a_{n+2}, a \times r^{(n+1)}=64}$

Dividing the second equation by the first, we get r2 = 4. Thus, r = 2 or -2 (both are valid in the context of GPs).

Application of Previous Concepts:

While determining the terms of a GP, sometimes you may need to first verify if the given sequence is indeed a GP. This could involve checking if the ratio between any two consecutive terms is constant (as introduced in Concept 1).

Tips and Tricks:

1. Consistent Ratio: Remember that for a sequence to be a GP, the ratio of any two consecutive terms should remain consistent.

2. Cross-multiplication trick: In problems where two terms of a GP are given, like in Question 1 above, often a quick cross-multiplication can help determine the common ratio.

3. Logarithmic Approach: In some complex problems or when dealing with higher powers of 'r', using logarithms can simplify the calculation.

4. Recognizing Powers: Often, terms in a GP may appear as powers of 2, 3, etc. Recognizing these can give instant clues about the first term and common ratio.

5. Real-world scenarios: In management exams, GP problems may be disguised in real-world situations, like compound interest, population growth, or depreciation. Being able to quickly recognize these scenarios as GP-related problems can save time.

6. Practice problems with negative ratios: Not all GPs increase or decrease. Some might oscillate in sign due to a negative common ratio. Practise these as they are a common twist in exams.

Remember, with GPs, it's essential to keep the nature of exponential growth or decay in mind. This helps not just in solving direct problems but also in making informed guesses or estimations when needed. Regular practice with a mix of straightforward and tricky problems ensures thorough preparation.