Asked by Kymnyque Ricks on Jun 04, 2024
Verified
Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. a1=10,a2=2a_{1}=10, a_{2}=2a1=10,a2=2
A) an=10(15) n−1a_{n}=10\left(\frac{1}{5}\right) ^{n-1}an=10(51) n−1
B) an=2n−1a_{n}=2^{n-1}an=2n−1
C) an=2(5) n−1a_{n}=2(5) ^{n-1}an=2(5) n−1
D) an=2(5) na_{n}=2(5) ^{n}an=2(5) n
E) an=10(15) xa_{n}=10\left(\frac{1}{5}\right) ^{x}an=10(51) x
Geometric Sequence
An ordered list of numbers in which every term succeeding the initial one is obtained by multiplying the preceding term by a constant, non-zero value known as the ratio.
Nth Term
The term located at position n in a sequence, where n represents a particular position number.
Formula
A formula is a mathematical equation or rule expressed in symbols.
- Apply formulas related to geometric sequences for the purpose of determining particular terms and summing subsets.
Verified Answer
JP
Jason PeterJun 10, 2024
Final Answer :
A
Explanation :
Since it is a geometric sequence, we know that the ratio between consecutive terms is constant. Let's call this ratio "r". Thus, we have:
a2a1=r⇒210=r⇒r=15\frac{a_{2}}{a_{1}} = r \Rightarrow \frac{2}{10} = r \Rightarrow r = \frac{1}{5}a1a2=r⇒102=r⇒r=51
Now we can use this ratio to find the n-th term:
an=a1rn−1=10(15)n−1a_{n} = a_{1}r^{n-1} = 10\left(\frac{1}{5}\right)^{n-1}an=a1rn−1=10(51)n−1
Therefore, choice A is the correct answer.
a2a1=r⇒210=r⇒r=15\frac{a_{2}}{a_{1}} = r \Rightarrow \frac{2}{10} = r \Rightarrow r = \frac{1}{5}a1a2=r⇒102=r⇒r=51
Now we can use this ratio to find the n-th term:
an=a1rn−1=10(15)n−1a_{n} = a_{1}r^{n-1} = 10\left(\frac{1}{5}\right)^{n-1}an=a1rn−1=10(51)n−1
Therefore, choice A is the correct answer.
Learning Objectives
- Apply formulas related to geometric sequences for the purpose of determining particular terms and summing subsets.