Problem 24 - Basic Engineering Mathematics

Question

In a geometric progression the sixth term is 8 times the third term and the sum of the seventh and eighth terms is 192 . Determine (a) the common ratio, (b) the first term, and (c) the sum of the fifth to eleventh terms, inclusive.

Step-by-Step Solution

Answer

(a) Common ratio is 2. (b) First term is 1. (c) Sum is 2032.

Step 1: Define the geometric progression formula

In a geometric progression, the nth term is given by \( a_n = ar^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio.

Step 2: Express the given terms in equation

We are told that the sixth term is 8 times the third term. The equation is \( ar^5 = 8(ar^2) \). The equation simplifies to \( r^3 = 8 \) because we can divide both sides by \( ar^2 \).

Step 3: Solve for the common ratio

From \( r^3 = 8 \), we find \( r = 2 \) because \( 2^3 = 8 \). Hence, the common ratio \( r \) is 2.

Step 4: Express the sum of terms as given

The sum of the seventh and eighth terms is 192, so we have:\( ar^6 + ar^7 = 192 \). Substituting \( r = 2 \), we get \( a(2^6) + a(2^7) = 192 \).

Step 5: Simplify and solve for the first term

Simplifying the equation \( a(64) + a(128) = 192 \) gives \( a(192) = 192 \).Thus, \( a = 1 \). Therefore, the first term \( a \) is 1.

Step 6: Determine the sum of the fifth to eleventh terms

The sum of terms from the fifth to eleventh in a geometric progression is given by the formula:\[ S = a \frac{r^m - r^n}{r-1} \]where \( m \) is the last term index and \( n \) is the first term index minus 1. For the fifth to eleventh terms:\( S = 1 \times \frac{2^{11} - 2^4}{2 - 1} = 2048 - 16 = 2032 \)

Key Concepts

Common Ratio

In a geometric progression, the **common ratio** is a crucial element that helps define the sequence. It is the factor by which we multiply a term to get the subsequent term. For example, if your sequence begins with the term \( a \) and your common ratio is \( r \), subsequent terms will be \( ar, ar^2, ar^3, \) and so on.
This pattern repeats indefinitely.In the given exercise, it is known that the sixth term is 8 times the third term. This tells us that:

• The sixth term \( ar^5 = 8 \times ar^2 \).
• After dividing each side by \( ar^2 \), we simplify this to \( r^3 = 8 \).

By solving \( r^3 = 8 \), we determine that \( r = 2 \). This means the terms in the sequence are doubled successively.

First Term

The **first term** in a geometric progression is the starting point of the sequence. This term, denoted as \( a \), sets the stage for all subsequent terms. For the sequence provided, we established the common ratio, but we also need the first term to fully describe the sequence.
The problem gives us that the sum of the seventh and eighth terms is 192. By substituting our previously found common ratio, we have:

• \( ar^6 + ar^7 = 192 \).
• Plug in \( r = 2 \), changing the equation to \( a\cdot64 + a\cdot128 = 192 \), simplifies the calculation.
• This follows to \( a(192) = 192 \), solving to \( a = 1 \).

Knowing \( a = 1 \) means the sequence starts with 1, and each subsequent term is obtained by multiplying by the common ratio of 2.

Sum of Terms

The **sum of terms** in a geometric sequence specific interval is calculated using a set formula.This formula calculates the sum \( S \) of a portion of the sequence from the \( n^{th} \) term to the \( m^{th} \) term:\[S = a \frac{r^m - r^n}{r-1}\]Where:

• \( a \) is the first term.
• \( r \) is the common ratio.
• \( m \) is the last term's index.
• \( n \) is the first term's index minus 1.

For our exercise, the sum from the fifth to the eleventh terms was requested:

• Apply the formula: \( S = 1 \times \frac{2^{11} - 2^4}{2 - 1} \).
• This simplifies to \( 2048 - 16 = 2032 \).

Hence, the sum of terms from the fifth to the eleventh is 2032.