Question

The same sequence may be defined in three ways: 1) f(x) = 3x + 4 for x = {1, 2, 3, ...}, 2) a1 = 4 and an + 1 = an + 3 for n = {1, 2, 3, ...} and 3) an = 4 + 3(n − 1) for n = {1, 2, 3, ...}.

Answer 1

The question involves defining the same arithmetic sequence through different mathematical formulas and relates to series and series expansions, like the binomial theorem, in high school level mathematics.

Step-by-step explanation:

The question deals with sequences and series in mathematics, where a sequence can be defined in multiple ways but yields the same result. In this particular case, a sequence is defined in three different ways, all describing the same arithmetic progression. The first way is a function f(x) = 3x + 4 for natural numbers x, the second is recurrently with a1 = 4 and an + 1 = an + 3 for indices n, and the third uses an explicit formula an = 4 + 3(n − 1).

Understanding sequences is crucial for grasping series expansions, such as the binomial theorem. Each term of the sequence becomes a part of the series when summed up. In the context provided, we can see how understanding powers and sequences can aid in comprehending series, particularly when dealing with the sums of arithmetic progressions or more complicated series expansions.