Question

what is the 7th term of the geometric sequence below?
7, -14, 28, -56, ...
a. -448
b. -896
c. 448
d. 896

Explanation:

Step1: Identify the first term and common ratio

The geometric sequence is \(7, -14, 28, -56, \dots\). The first term \(a_1 = 7\). The common ratio \(r\) is found by dividing a term by its previous term, e.g., \(r=\frac{-14}{7}=-2\).

Step2: Use the geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n - 1}\). We need the 7th term, so \(n = 7\). Substitute \(a_1 = 7\), \(r=-2\), and \(n = 7\) into the formula:
\(a_7 = 7 \cdot (-2)^{7 - 1}\)

Step3: Simplify the exponent and calculate

First, simplify the exponent: \(7 - 1 = 6\). Then calculate \((-2)^6 = 64\). Now multiply by 7: \(a_7 = 7 \cdot 64 = 448\)? Wait, no, wait. Wait, the sign: \((-2)^6\) is positive 64, but wait, let's check the sequence. The terms alternate sign. Wait, \(n = 7\), so \(n - 1 = 6\), which is even, so \((-2)^6 = 64\), so \(7 \times 64 = 448\)? But wait, the 4th term is -56, 5th term would be \(-56 \times (-2)=112\), 6th term \(112 \times (-2)= -224\), 7th term \(-224 \times (-2)=448\)? Wait, no, wait my mistake earlier. Wait, let's recalculate step by step.

Wait, 1st term: 7

2nd term: 7 * (-2) = -14

3rd term: -14 * (-2) = 28

4th term: 28 * (-2) = -56

5th term: -56 * (-2) = 112

6th term: 112 * (-2) = -224

7th term: -224 (-2) = 448. Wait, but the options have C as 448. Wait, but let's use the formula again. \(a_n = a_1 r^{n - 1}\). So \(a_7 = 7 \times (-2)^{6}\). \((-2)^6 = 64\), 764=448. So the 7th term is 448. Wait, but let's check the options. Option C is 448. Wait, but wait, did I make a mistake in the sign? Wait, the common ratio is -2, so each term is previous term times -2. So 5th term: -56 (-2)=112 (positive), 6th term: 112 (-2)= -224 (negative), 7th term: -224 * (-2)=448 (positive). Yes, that's correct. So the 7th term is 448.

Wait, but let's re-express the formula. \(a_n = a_1 r^{n - 1}\). So \(a_7 = 7 \times (-2)^{6}\). \((-2)^6 = 64\), 7*64=448. So the answer is C.

Answer:

C. 448