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.