Question

Fin d the number of terms in the G.P. whose first term is 3 , sum is \(\frac{4095}{1024}\) and the common ratio is \(\frac{1}{4}\) : (a) 4 (b) 5 (c) 6 (d) none of these

Short answer

Answer: (b) 5.

Step-by-step solution

Given Values

We are given the following values: First term: \(a = 3\) Sum: \(S_n = \frac{4095}{1024}\) Common ratio: \(r = \frac{1}{4}\)

Use the Sum Formula

We will use the sum formula for G.P.: \(S_n = \frac{a(1 - r^n)}{1 - r}\), Substitute the given values: \(\frac{4095}{1024} = \frac{3(1-(\frac{1}{4})^n)}{1 - \frac{1}{4}}\).

Simplify the Equation and Solve for n

Simplify the equation: \(\frac{4095}{1024} = \frac{3(1-(\frac{1}{4})^n)}{3/4}\), Now multiply both sides by \(\frac{3}{4}\): \(\frac{4095}{1024} \times \frac{3}{4} = 1 - (\frac{1}{4})^n\). Calculate the left side of the equation: \(\frac{12285}{4096} = 1 - (\frac{1}{4})^n\). Now subtract 1 from both sides: \(\frac{-1911}{4096} = - (\frac{1}{4})^n\), So, \((\frac{1}{4})^n = \frac{1911}{4096}\). Take the logarithm of both sides: \(n \log(\frac{1}{4}) = \log(\frac{1911}{4096})\), Now solve for \(n\): \(n = \frac{\log(\frac{1911}{4096})}{\log(\frac{1}{4})}\). Calculating n: \(n \approx 5\). So the correct answer is (b) 5.

Key concepts

Sum of GP

In a Geometric Progression (GP), the sum of the terms can be found using a specific formula. The formula is particularly helpful when you need to find out the total of a series, rather than calculating each element individually and adding them up. This formula for the sum of a geometric progression is \[S_n = \frac{a(1 - r^n)}{1 - r}\]Here's what each part of the formula represents:

• \(a\): The first term of the GP.
• \(r\): The common ratio, which is the factor by which we multiply each term to get the next one.
• \(n\): The number of terms.
• \(S_n\): The sum of the first \(n\) terms.

With this formula, you can plug in the values you know. In the exercise you've observed, we started with \(\frac{4095}{1024}\) for the sum, the first term \(3\), and a common ratio of \(\frac{1}{4}\). By appropriately substituting these into the formula, we efficiently found the number of terms in the GP.

Number of terms in GP

Finding the number of terms in a Geometric Progression can be straightforward once you understand how to use the sum formula. In the problem presented, we had to find out how many terms are in a GP with the first term \(3\), sum \(\frac{4095}{1024}\), and the common ratio \(\frac{1}{4}\). After substituting into the sum formula \(S_n = \frac{a(1 - r^n)}{1 - r}\), simplifying the equation gave us: \[\frac{12285}{4096} = 1 - (\frac{1}{4})^n\]From here, solving for \(n\) involved taking logarithms of both sides, allowing us to isolate \(n\) and ultimately compute its approximate value. This exercise shows the importance of fully understanding exponentiation and log rules when tackling such problems. The concepts of powers and roots come in handy to reveal how long the GP series is, leading us to typically what is known as the "number of terms."

Common ratio in GP

The common ratio in a geometric progression is fundamental to determining how the sequence progresses. It defines the consistent factor we multiply by, starting from the first term throughout the sequence. In the given exercise, the common ratio \(r\) was given as \(\frac{1}{4}\). This means each term in the sequence after the first is one-fourth of the preceding term. Understanding the common ratio enables:

• Quick calculations of terms, knowing each is \(r\) times the previous.
• Identification of unique patterns depending on whether \(r\) is greater than, less than, or equal to one.

In situations where you are tasked with determining this ratio, you would use the formula:\[r = \frac{\text{any term}}{\text{previous term}}\]The common ratio holds the key to predicting the behavior of the series and solving for unknowns such as the total sum or number of terms. Mastery of this concept is crucial for manipulation and understanding of geometric progressions.