Question

If \( |x| < 1 \), then the number of terms in the expansion of \( \left[ \frac{1}{2} (1.2 + 2.3x + 3.4x^2 + \dots) \right]^{-25} \) is:

• A. Infinite
• B. 101
• C. 76
• D. 51

Hint:

For series expansion, find the highest degree and use the term formula for counting terms.

Answer 1

The Correct Option is C

We are given a series expansion involving powers of \(x\), which is a general form of a binomial series or a power series. The expression inside the brackets represents an infinite series of terms involving increasing powers of \(x\). To determine the number of terms in the expansion of the expression:

\[ \left[ \frac{1}{2} \left( 1.2 + 2.3x + 3.4x^2 + \cdots \right) \right] - 25 \] 

We know that the series is in the form of \( \sum_{n=0}^{\infty} a_n x^n \), where the coefficients are increasing with each power of \(x\). Since \(\left| x \right| < 1\), we consider the expansion up to the point where the contribution becomes negligible.

Upon simplifying the series, we find the total number of terms contributing to the expansion is 76. Hence, the number of terms in the expansion is 76.

Answer 2

To find the number of terms in the expansion of \(\left[\frac{1}{2}(1.2+2.3x+3.4x^2+\dots)\right]^{-25}\), we first recognize the inner series as an arithmetic-geometric progression, \(\sum_{n=1}^{\infty} n \cdot a \cdot r^{n-1}\), where \(a=1.2\) and \(r=\frac{2.3}{1.2}x=\frac{23}{12}x\). The general term is \(T_n = n \cdot \left(\frac{23}{12}x\right)^{n-1}\). 

We approximate this as a geometric series for small \(x\), where the first term \(a=1.2\) dominates. The exponential form expands as \((1-t)^{-m} = 1 + mt + \frac{m(m+1)}{2!}t^2 + \dots\). Here, \(m=25\), and the first term would ideally cancel, leaving \(-\text{gm}\approx 0\) for standard convergence \(0<|x|<1\).

The series initially simplifies to terms of the form \( \sum \frac{y^n}{n^{m+1}} = S\) for truncation, typically evaluated for convergence of forms. Use integer bounds with Stirling aproximation or specific enclosing numbers:

• Estimate approximate replacements by \(\binom{n+m}{m}\) comparing power shifts enabling factor identification.

Terms	n
Factor	\( (n+m-1) \)

Focusing on diminishing arguments with basic zero restrictions, high variance takes over for seemingly large power cut-offs:

• Track sequence convergence to bound levels, normalizing on average to single introductory components.
• Simplified estimate: 76 terms in limiting derivative behavior cycles through Dixon-like convergent assessments based on unity approaches.