Question

The number of terms in the expansion of $(1 + x)(1 + {x^3})(1 + {x^6})(1 + {x^{12}})(1 + {x^{24}}) \cdot \cdot \cdot \cdot (1 + {x^{3 \times {2^n}}})$ is:
A. ${2^{n + 3}}$
B. ${2^{n + 4}}$
C. ${2^{n + 5}}$
D. None of these

Answer 1

Hint: This a problem related to the algebraic expressions and algebraic equations, though this problem seems complex but it is not, it simply deals with the sequence uniformity and how the number of terms are progressing as it is by multiplied with more terms. This deals with observing how the expansion is proceeding forward, by observing that we generalize a formula to solve the problem.

Complete step by step answer:
Consider the first two expressions:
$ \Rightarrow (1 + x)(1 + {x^3})$
Now expand the considered first two expressions:
\[ \Rightarrow (1 + x)(1 + {x^3}) = 1 + x + {x^3} + {x^4}\]
On expansion there are 4 terms, where $4 = 2 \times 2;$
Now consider the next expression along with the expanded expression:
$ \Rightarrow (1 + x + {x^3} + {x^4})(1 + {x^6}) = 1 + x + {x^3} + {x^4} + {x^6} + {x^7} + {x^9} + {x^{10}}$
Here on expansion , there are 8 terms, where $8 = {2^2} \times 2;$ when multiplied with $(1 + {x^6})$, here $6 = 3 \times {2^1};$
Now considering the next term along with the previous expanded expression:
$ \Rightarrow (1 + x + {x^3} + {x^4} + {x^6} + {x^7} + {x^9} + {x^{10}})(1 + {x^{12}})$, on multiplication gives:
$ \Rightarrow 1 + x + {x^3} + {x^4} + {x^6} + {x^7} + {x^9} + {x^{12}} + {x^{13}} + {x^{15}} + {x^{19}} + {x^{21}} + {x^{22}}$
On further expansion, there are 16 terms, where $16 = {2^3} \times 2;$ when multiplied with $(1 + {x^{12}})$, here $12 = 3 \times {2^2};$
$\therefore $There is a pattern in which the terms are expanding in the form of ${2^n} \times 2;$ whenever it multiplied with a polynomial of degree $3 \times {2^{n - 1}};$
So here the last polynomial is of degree $3 \times {2^n}$; hence the number of terms after expanding should be equal to ${2^{n + 1}} \times 2 = {2^{n + 2}};$
$\therefore $The number of terms in the expansion of $(1 + x)(1 + {x^3})(1 + {x^6})(1 + {x^{12}})(1 + {x^{24}}) \cdot \cdot \cdot \cdot (1 + {x^{3 \times {2^n}}})$ is ${2^{n + 2}}$.

So, the correct answer is “Option D”.

Note: Every time expanding any sequence carefully observe if it is following any pattern, and then solve accordingly. Here first started observing with these first two polynomials and then continued the observation throughout the expansion and detected a pattern, or a generalized formula in expanding the series and applied the formula to the sequence to get the total number of terms.