Question

Find the common ratio of the G.P. whose first and last terms are 5 and \(\frac{32}{625}\) respectively and the sum of the G.P. is \(\frac{5187}{625}\) : (a) \(\frac{1}{5}\) (b) \(\frac{2}{5}\) (c) \(\frac{5}{3}\) (d) \(\frac{4}{5}\)

Short answer

Answer: The common ratio of the geometric progression is \(\frac{2}{5}\).

Step-by-step solution

Write down the given values and the unknowns

The given values are: First term (\(a_1\)) = 5 Last term (\(T_n\)) = \(\frac{32}{625}\) Sum of the G.P. (\(S_n\)) = \(\frac{5187}{625}\) The unknown values that we want to find are the number of terms (n) and the common ratio (r).

Use the formula for the nth term of a G.P. to express the last term in terms of n and r

We know that \(T_n = a_1 \cdot r^{n-1}\). Substituting the given values into this equation, we get: \(\frac{32}{625} = 5 \cdot r^{n-1}\)

Solve for the common ratio (r) in terms of n

To solve the equation from Step 2 for r, we will need to isolate r: \(r^{n-1} = \frac{32}{625 \cdot 5} = \frac{32}{3125}\) Since we want to isolate r, we will take the nth root of the equation: \(r = \sqrt[n]{\frac{32}{3125}}\)

Use the formula for the sum of a G.P. to express the sum in terms of n and r

We know that \(S_n = \frac{a_1(r^n - 1)}{r - 1}\). Substituting the given values and the expression for r from Step 3, we get: \(\frac{5187}{625} = \frac{5(\sqrt[n]{\frac{32}{3125}}^n - 1)}{\sqrt[n]{\frac{32}{3125}} - 1}\)

Solve the equation from Step 4 to find the common ratio (r)

Observing the options for r, we see they are all in fraction form. To find which option fits, we substitute each option into the equation from Step 4 and simplify: (a) If r = \(\frac{1}{5}\), the equation becomes: \(\frac{5187}{625} = \frac{5(\frac{1}{5}^n - 1)}{\frac{1}{5} - 1}\) This option doesn't lead to a whole integer value for n, so it's not the correct option. (b) If r = \(\frac{2}{5}\), the equation becomes: \(\frac{5187}{625} = \frac{5(\frac{2}{5}^n - 1)}{\frac{2}{5} - 1}\) Solving this equation, we find n = 6, which is a whole number. Thus, the correct option is r = \(\frac{2}{5}\).

Key concepts

Geometric Progression (G.P.)

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding this concept is crucial for problems involving G.P.s, as it builds the foundation for finding terms and sums within the sequence.

For example, in the sequence 2, 6, 18, 54, the common ratio is 3 because each term is three times the previous one. Knowing the common ratio allows us to predict any term in the sequence and calculate the sum of a certain number of terms. Always remember that in a G.P., the common ratio remains constant, that is, each term is the result of multiplying the preceding term by the same ratio.

Sum of Geometric Progression

The sum of a geometric progression can be calculated using a specific formula, and it's a common exercise in algebra. If we have a G.P. with a first term of 'a', a common ratio of 'r', and 'n' number of terms, the sum of the first 'n' terms is given by:

\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
if 'r' is not equal to 1.

When the Common Ratio is Less than One

If the common ratio is less than one (but greater than zero), the terms of the G.P. get progressively smaller. As 'n' gets very large, the terms approach zero, and the sum approaches a finite limit.

When the Common Ratio is Greater than One

Conversely, if the common ratio is greater than one, the terms get larger and the sum can grow without bound as 'n' increases. It's important to know how to manipulate this formula not only to find the sum but also to infer properties like the number of terms or the common ratio when other information is given.

nth Term of a Geometric Progression

The nth term of a geometric progression is key in understanding how to navigate through any G.P. exercise. To find the nth or any specific term in a geometric progression, we use the formula:

\[ T_n = a \times r^{n-1} \]
where 'T_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the sequence. By substituting the known values into this equation, you can solve for any unknown variable. For instance, if a problem gives you the first term, the common ratio, and the position of the term you're looking for, you can directly calculate the value of that term.

Root-Taking in Algebra

Root-taking is an algebraic operation used to find the original number that was raised to a power to get a certain number. In the context of a G.P., finding a common ratio often requires root-taking, especially when dealing with fractional exponents or solving for a variable raised to the power of another variable.

For example, to isolate 'r' in the equation \( r^n = a \), you would take the nth root of both sides, which gives \( r = \sqrt[n]{a} \). This operation is essential when you're faced with determining unknowns in an equation that represents a geometric series, as you often need to untangle variables from their exponents. Mastery of root-taking, along with exponentiation, is vital for students tackling algebraic problems involving geometric progressions.