Series

In a geometric progression (GP), each term after the first is found by multiplying the previous term by a constant called the common ratio (r). The formula for the sum of the first $n$ terms of a geometric progression is:

$S_n = A_1 \cdot \frac{1 - r^n}{1 - r} \quad \text{for} \quad r \neq 1$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $A_1$ is the first term.
• $r$ is the common ratio.
• $n$ is the number of terms.

Example 4: Find the sum of the geometric series 2 + 4 + 8 + 16 + ……………….. + 2048

In this case, $A_1 = 2$, $r = 2$, and $A_n = 2048$. We need to find $n$. Using the formula for the nth term of a geometric progression:

$A_n = A_1 \cdot r^{n-1}$

Substitute the known values into the formula:

$2048 = 2 \cdot 2^{n-1}$

$2048 = 2^n$

Taking the logarithm base 2 of both sides:

$n = \log_2{2048} = 11$

The number of terms is $n = 11$. Now, use the sum formula for a geometric series:

$S_n = A_1 \cdot \frac{1 - r^n}{1 - r}$

$S_{11} = 2 \cdot \frac{1 - 2^{11}}{1 - 2} = 2 \cdot \frac{1 - 2048}{-1} = 2 \cdot (2047)$

$S_{11} = 4094$

The sum of the first 11 terms is $S_{11} = 4094$.

Example 5: Find the sum of the first 8 terms of the series 5 + 20 + 80 + 320 + …………..

In this case, $A_1 = 5$, $r = 4$, and $n = 8$. Using the formula for the sum of the first $n$ terms of a geometric progression:

$S_8 = 5 \cdot \frac{1 - 4^8}{1 - 4}$

$S_8 = 5 \cdot \frac{1 - 65536}{-3} = 5 \cdot \frac{-65535}{-3} = 5 \cdot 21845 = 109225$

The sum of the first 8 terms is $S_8 = 109225$.

The geometric mean (GM) of $n$ positive numbers is the nth root of their product. For example, the geometric mean of $a$, $b$, $c$, and $d$ is:

$GM = \sqrt[n]{a \cdot b \cdot c \cdot d}$

• If $a$, $b$, and $c$ are consecutive terms of an arithmetic progression (A.P.), then $b$ is the arithmetic mean of $a$ and $c$.
• If $a$, $b$, and $c$ are consecutive terms of a geometric progression (G.P.), then $b$ is the geometric mean of $a$ and $c$.

Consider three terms a, b, and c in an arithmetic progression. By the definition of an A.P., we know that:

b - a = c - b

This implies:

2b = a + c

Therefore, the arithmetic mean (A.M.) of a and c is:

A.M. = (a + c) / 2 = b

Conclusion: In an A.P., the middle term is the arithmetic mean of the other two terms.

Consider three terms a, b, and c in a geometric progression. By the definition of a G.P., we know that:

b / a = c / b

This implies:

b^2 = a * c

Therefore, the geometric mean (G.M.) of a and c is:

G.M. = √(a * c) = b

Conclusion: In a G.P., the middle term is the geometric mean of the other two terms.

Example 6: Find the arithmetic and geometric means of the following sequence:

Given the sequence 3, 6, and 12:

• The arithmetic mean is $\frac{3 + 12}{2} = 7.5$.
• The geometric mean is $\sqrt{3 \times 12} = \sqrt{36} = 6$.