Question

khalid wrote the sequence below.
x−1, −2x+2, 4x−4, −8x+8, …
which formula can be used to find the 10th term of the sequence?
a₁₀ = (x−1)(−2)¹⁰⁻¹
a₁₀ = (x−1)¹⁰⁻¹
a₁₀ = (x−1)(2)¹⁰⁻¹
a₁₀ = (x−1)(−2)¹⁰

Explanation:

Step1: Identify the sequence type

The given sequence is \( x - 1, -2x + 2, 4x - 4, -8x + 8, \dots \). Let's check the common ratio \( r \) between consecutive terms.
For a geometric sequence, \( r=\frac{a_{n + 1}}{a_n}\).
First, factor the terms:

• \( a_1=x - 1\)
• \( a_2=-2x + 2=-2(x - 1)\)
• \( a_3=4x - 4 = 4(x - 1)=(-2)^2(x - 1)\)
• \( a_4=-8x + 8=-8(x - 1)=(-2)^3(x - 1)\)

So, the common ratio \( r=-2\) and the first term \( a_1=(x - 1)\).

Step2: Recall the formula for the nth term of a geometric sequence

The formula for the \( n \)-th term of a geometric sequence is \( a_n=a_1\times r^{n - 1}\).

Step3: Find the 10th term

Here, \( n = 10\), \( a_1=(x - 1)\), and \( r=-2\). Substitute these values into the formula:
\( a_{10}=a_1\times r^{10 - 1}=(x - 1)\times(-2)^{10 - 1}\)

Answer:

\( a_{10}=(x - 1)(-2)^{10 - 1} \) (the first option: \( a_{10}=(x - 1)(-2)^{10 - 1} \))