The graph represents the sequence is Option D.
Further explanation
A function defined in the set of natural numbers is called a sequence.
Allow [tex]\boxed{a_n \ as \ n^{th} \ term} [/tex], or general term.
In a sequence, n should always represent a natural number, i.e.,
n > 0, n = 1, 2, 3, ...,
but the value of [tex]\boxed{a_n} [/tex] may be any real number depending on the formula for the general term of the sequence.
A sequence is considered geometric if the ratio between each consecutive term is common.
In our problem, the sequence is [tex]\boxed{ \ \frac{1}{2}, 1, 2, 4, 8, ... \ }[/tex]
The ratio of each term [tex] \boxed{a_{n+1}} [/tex] to the previous term [tex]\boxed{a_n} [/tex] is equal 2, so we can formalize the sequence as
[tex] \boxed{\frac{a_{n+1}}{a_n} =2}. [/tex]
The consecutive terms of the sequence have a common ratio r = 2, so this sequence is geometric.
The general term of a geometric sequence [tex]\boxed{a_n} [/tex] with common ratio r is [tex] \boxed{\boxed{ \ a_n = a_1 \cdot r^{n-1} \ }}. [/tex]
Presently we go back to the question. The graph shows the horizontal axis as n and the vertical axis is the general term [tex]\boxed{a_n} [/tex]. The relationship between n, the terms, and the coordinates as written below:
[tex]\boxed{n = 1 \rightarrow the \ 1st \ term \ a_1 = \frac{1}{2} \rightarrow \bigg( 1, \frac{1}{2} \bigg)}[/tex]
[tex]\boxed{n = 2 \rightarrow the \ 2nd \ term \ a_2 = 1 \rightarrow (2, 1)}[/tex]
[tex]\boxed{n = 3 \rightarrow the \ 3rd \ term \ a_3 = 2 \rightarrow (3, 2)}[/tex]
[tex]\boxed{n = 4 \rightarrow the \ 4th \ term \ a_4 = 4 \rightarrow (4, 4)}[/tex]
[tex]\boxed{n = 5 \rightarrow the \ 5th \ term \ a_5 = 8 \rightarrow (5, 8)}[/tex]
Therefore, the graph representing the sequence is Option D.
Note:
- The general term of a geometric sequence is exponential.
- From the common ratio (r > 1) and graph, the type is an increasing sequence.
Learn more
- Combining two functions to create a geometric sequence https://brainly.com/question/1695742
- A word problem about arithmetic and geometric sequences https://brainly.com/question/3395975
- Drawing graph of the geometric sequence https://brainly.com/question/3166290
Keywords: which, the graph, geometric sequences, common ratio, general term formula, natural numbers, The consecutive terms, arithmetic
