Question

1. 1 is the first term in the geometric sequence 1, -4, 16, -64, .... what is the seventh term of the sequence?

f -1024
g -76
h 256
j 4096

Explanation:

Step1: Identify the formula for geometric sequence

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \times r^{(n - 1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step2: Find the common ratio \(r\)

To find \(r\), divide the second term by the first term: \(r=\frac{-4}{1}=-4\).

Step3: Substitute values into the formula for the 7th term

Here, \(a_1 = 1\), \(r=-4\), and \(n = 7\). So, \(a_7=1\times(-4)^{(7 - 1)}\).

Step4: Calculate \((-4)^{6}\)

\((-4)^{6}=(-4)\times(-4)\times(-4)\times(-4)\times(-4)\times(-4)=4096\)? Wait, no, wait: Wait, \((-4)^6\) is positive because the exponent is even. But wait, let's check the sequence: 1, -4, 16, -64,... The sign alternates. Wait, \(n = 7\), so \(n-1 = 6\), which is even. Wait, but the first term is 1 (term 1, \(n=1\)), term 2: \(1\times(-4)=-4\) (n=2), term 3: \(-4\times(-4)=16\) (n=3), term 4: \(16\times(-4)=-64\) (n=4), term 5: \(-64\times(-4)=256\) (n=5), term 6: \(256\times(-4)=-1024\) (n=6), term 7: \(-1024\times(-4)=4096\)? Wait, no, wait my mistake earlier. Wait, let's recalculate:

Wait, \(a_n=a_1\times r^{n - 1}\). So for \(n = 7\), \(a_7=1\times(-4)^{6}\). \((-4)^6 = 4096\)? But wait the options have J as 4096. Wait but let's check the sequence again. Wait term 1: 1, term 2: -4 (1(-4)), term 3: 16 ( -4(-4)), term 4: -64 (16(-4)), term 5: 256 (-64(-4)), term 6: -1024 (256(-4)), term 7: 4096 (-1024(-4)). Yes, so term 7 is 4096? But wait the option F is -1024 (which is term 6), G is -76 (not a geometric term), H is 256 (term 5), J is 4096 (term 7). Wait, but wait, did I make a mistake in the sign? Wait, \(r=-4\), so \(a_n = a_1r^{n - 1}\). So for \(n = 7\), \(r^{6}=(-4)^6 = 4096\), so \(a_7 = 1\times4096 = 4096\). So the 7th term is 4096, which is option J. Wait, but let's check again:

Term 1: 1 (n=1)

Term 2: 1*(-4) = -4 (n=2)

Term 3: -4*(-4) = 16 (n=3)

Term 4: 16*(-4) = -64 (n=4)

Term 5: -64*(-4) = 256 (n=5)

Term 6: 256*(-4) = -1024 (n=6)

Term 7: -1024*(-4) = 4096 (n=7)

Yes, so term 7 is 4096, which is option J. Wait, but earlier I thought maybe I messed up, but no, that's correct.

Wait, but wait the options: F is -1024 (term 6), J is 4096 (term 7). So the correct answer is J. 4096.

Wait, but let's re - express the formula:

\(a_n=a_1r^{n - 1}\)

\(a_1 = 1\), \(r=-4\), \(n = 7\)

\(a_7=1\times(-4)^{6}\)

\((-4)^6 = (-4)\times(-4)\times(-4)\times(-4)\times(-4)\times(-4)=4096\) (since 6 is even, the result is positive)

So the 7th term is 4096.

Answer:

J. 4096