Question

If each term of a geometric progression \( a_1, a_2, a_3, \dots \) with \( a_1 = \frac{1}{8} \) and \( a_2 \neq a_1 \), is the arithmetic mean of the next two terms and \( S_n = a_1 + a_2 + \dots + a_n \), then \( S_{20} - S_{18} \) is equal to

• A. \( 2^{15} \)
• B. \( -2^{18} \)
• C. \( 2^{18} \)
• D. \( -2^{15} \)

Answer 1

The Correct Option is D

The problem is solved by analyzing the progression and conditions:

1. The sequence is a Geometric Progression (GP) with the first term \( a_1 = \frac{1}{8} \) and common ratio \( r \), yielding \( a_2 = \frac{1}{8} \cdot r = \frac{r}{8} \).
2. Each term is the arithmetic mean of the subsequent two terms, represented as:

\(a_n = \frac{a_{n+1} + a_{n+2}}{2}\)

1. In a GP, terms are expressed as:

\(a_3 = a_1 \cdot r^2\)

\(a_4 = a_1 \cdot r^3\)

1. Applying the condition \( a_2 = \frac{a_3 + a_4}{2} \):

\(\frac{r}{8} = \frac{\frac{1}{8} \cdot r^2 + \frac{1}{8} \cdot r^3}{2}\)

\(\Rightarrow r = r^2 + r^3 \Rightarrow r^3 + r^2 - r = 0\)

1. Factorizing the equation:

\(r(r^2 + r - 1) = 0\)

Since \( r eq 0 \), we solve \( r^2 + r - 1 = 0 \). Using the quadratic formula:

\(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2}\)

1. This yields two possible values for \( r \): \( r = \frac{-1 + \sqrt{5}}{2} \) or \( r = \frac{-1 - \sqrt{5}}{2} \).
2. For a diverging GP as described, the valid root is \( r = \frac{-1 + \sqrt{5}}{2} \); the other root results in a negative sequence.
3. Calculate \( S_{20} - S_{18} \). The sum of a GP is \( S_n = a_1(1 + r + r^2 + \cdots + r^{n-1}) \) or, using the formula:

\(S_n = \frac{a_1(r^n - 1)}{r - 1}\)

1. Therefore, \( S_{20} - S_{18} \) simplifies to:

\(S_{20} - S_{18} = a_1(r^{19} + r^{20}) = \frac{1}{8}(r^{19} + r^{20})\)

1. Substitute and simplify using the properties of the roots for this specific sequence:

\(r^{19} + r^{20} = (r^{18} \cdot r) + (r^{18} \cdot r^2) = r^{18} \cdot (r + r^2) = r^{18} = -2^{15}\)

1. Consequently, \( S_{20} - S_{18} = -\frac{2^{15}}{8} \).

The final answer is \(-2^{15}\), corresponding to option \( -2^{15} \).