A sequence has the following terms: $$ a_{1}=4, a_{2}=10, a_{3}=25, a_{4}=62.5 $$. Which formula represents the $$ n^{t h} $$ term in this sequence? A. $$ a_{n}=4+2.5 n $$ B. $$ a_{n}=4+2.5(n-1) $$ C. $$ a_{n}=4(2.5)^{n} $$ D. $$ a_{n}=4(2.5)^{n-1} $$

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UpStudy AI · Accepted Answer

We start by recognizing that the ratio between successive terms is constant. Compute the ratios:

$$
\frac{a_2}{a_1} = \frac{10}{4} = 2.5, \quad \frac{a_3}{a_2} = \frac{25}{10} = 2.5, \quad \frac{a_4}{a_3} = \frac{62.5}{25} = 2.5
$$

Since the ratio is $$2.5$$ for each step, the sequence is geometric with common ratio $$r = 2.5$$. The general formula for the $$ n^{\text{th}} $$ term of a geometric sequence is:

$$
a_{n} = a_{1} \cdot r^{n-1}
$$

Substitute $$ a_{1} = 4 $$ and $$ r = 2.5 $$:

$$
a_{n} = 4 \cdot (2.5)^{n-1}
$$

Thus, the formula that represents the $$ n^{\text{th}} $$ term in this sequence is:

$$
a_{n} = 4(2.5)^{n-1}
$$
The formula for the $$ n^{\text{th}} $$ term is $$ a_{n} = 4(2.5)^{n-1} $$, which is option D.

A sequence has the following terms: \( a_{1}=4, a_{2}=10, a_{3}=25, a_{4}=62.5 \). Which formula represents the \( n^{t h} \) term in this sequence? A. \( a_{n}=4+2.5 n \) B. \( a_{n}=4+2.5(n-1) \) C. \( a_{n}=4(2.5)^{n} \) D. \( a_{n}=4(2.5)^{n-1} \)

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