Final answer:
The question is about finding a term in a cumulative sum sequence in mathematics. It discusses manipulating a series to prove an identity, similar to the Binomial theorem, and reasoning about sums using comparisons of fractions.
Step-by-step explanation:
The question involves a sequence in mathematics where each term is the sum of all the previous terms. We need to find the value of the first term which exceeds a certain number. This type of sequence is known as a cumulative or partial sum sequence. In this particular case, the sequence displays a property that can be explained using a series expansion or manipulation akin to what is used in proving mathematical identities such as the Binomial theorem.
For n terms in the sequence, we are given that the expression is equal to n². By manipulating the terms of the series (taking (n - 1) from the last term and adding it to the first term, etc.), we can simplify the expression to 2n². This is a mathematical trick that is used to identify the sum of a particular series.
The series expansions mentioned, like the Binomial theorem, show how sequences and series can be expanded or summed in a systematic way using algebraic techniques. Lastly, the point about 1 + ¼ > 1 and that one-third is smaller than one-half indicates a basic understanding of comparing fractions and using logical reasoning to bound sums of series.