Question

given the geometric sequence 8, 10, 12.5, 15.65, ..., which of the following are true? ( select all that apply )
show your work here
□ $a_n = 1.25(8)^{n - 1}$ □ $a_n = 8(1.25)^{n - 1}$
□ $a_n = 6.4(1.25)^n$ □ $a_n = 1.25(a_n - 1); a_1 = 8$
□ $s_{10} = 59.6$ □ $s_{10} = 266.02$

Explanation:

Step1: Find the common ratio \( r \)

In a geometric sequence, \( r=\frac{a_{n}}{a_{n - 1}} \). For the given sequence \( 8,10,12.5,15.625,\dots \), \( r=\frac{10}{8} = 1.25\), \( \frac{12.5}{10}=1.25 \), so \( r = 1.25 \), and the first term \( a_{1}=8 \).

Step2: Recall the geometric sequence formula

The explicit formula for a geometric sequence is \( a_{n}=a_{1}(r)^{n - 1} \). Substituting \( a_{1}=8 \) and \( r = 1.25 \), we get \( a_{n}=8(1.25)^{n - 1} \), so this option is correct. The option \( a_{n}=1.25(8)^{n - 1} \) has the wrong first term and ratio placement, so it's incorrect.

Step3: Check the recursive formula

A geometric sequence has the recursive formula \( a_{n}=r\times a_{n - 1} \) with \( a_{1} \) as the first term. Here, \( r = 1.25 \) and \( a_{1}=8 \), so \( a_{n}=1.25(a_{n - 1}) \) with \( a_{1}=8 \), so this option is correct.

Step4: Check the formula \( a_{n}=6.4(1.25)^{n} \)

Let's test \( n = 1 \): \( 6.4(1.25)^{1}=6.4\times1.25 = 8 \), which matches \( a_{1} \). For \( n = 2 \): \( 6.4(1.25)^{2}=6.4\times1.5625 = 10 \), which matches \( a_{2} \). For \( n = 3 \): \( 6.4(1.25)^{3}=6.4\times1.953125 = 12.5 \), which matches \( a_{3} \). Let's derive it from the explicit formula: \( a_{n}=8(1.25)^{n - 1}=\frac{8}{1.25}(1.25)^{n}=6.4(1.25)^{n} \), so this option is correct.

Step5: Calculate the sum of the first 10 terms \( S_{10} \)

The sum of the first \( n \) terms of a geometric sequence is \( S_{n}=\frac{a_{1}(r^{n}-1)}{r - 1} \) (since \( r>1 \)). Substituting \( a_{1}=8 \), \( r = 1.25 \), and \( n = 10 \):

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So \( S_{10}\approx266.02 \), so the option \( S_{10}=266.02 \) is correct, and \( S_{10}=59.6 \) is incorrect.

Answer:

The correct options are:

• \( a_{n} = 8(1.25)^{n - 1} \)
• \( a_{n} = 1.25(a_{n} - 1); a_{1} = 8 \) (Note: The notation here should be \( a_{n}=1.25(a_{n - 1}); a_{1}=8 \), assuming a typo in the original, but the meaning is clear)
• \( a_{n} = 6.4(1.25)^{n} \)
• \( S_{10} = 266.02 \)

(In boxed form for each correct option as per the question's multiple - select requirement, but since we need to list them: )
\( \boldsymbol{a_{n} = 8(1.25)^{n - 1}} \), \( \boldsymbol{a_{n} = 1.25(a_{n - 1}); a_{1} = 8} \), \( \boldsymbol{a_{n} = 6.4(1.25)^{n}} \), \( \boldsymbol{S_{10} = 266.02} \)