Math Problem Statement

Given a geometric sequence whose 1st term = 48 and 4th term = 384.

Find the 7th term.

Solution

We are given a geometric sequence with the following information:

  • The 1st term (a1a_1) = 48
  • The 4th term (a4a_4) = 384

To solve for the 7th term, we will use the formula for the nn-th term of a geometric sequence:

an=a1rn1a_n = a_1 \cdot r^{n-1} Where:

  • ana_n is the nn-th term
  • a1a_1 is the first term
  • rr is the common ratio
  • nn is the term number

Step 1: Use the formula to express the 4th term

For the 4th term:

a4=a1r41=a1r3a_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3

Substitute the known values:

384=48r3384 = 48 \cdot r^3

Step 2: Solve for rr

Divide both sides by 48:

r3=38448=8r^3 = \frac{384}{48} = 8

Now take the cube root of both sides:

r=83=2r = \sqrt[3]{8} = 2

Step 3: Find the 7th term

Now that we know r=2r = 2, we can find the 7th term (a7a_7) using the formula:

a7=a1r71=4826a_7 = a_1 \cdot r^{7-1} = 48 \cdot 2^6

Calculate 262^6:

a7=4864=3072a_7 = 48 \cdot 64 = 3072

Thus, the 7th term is 3072.


Would you like more details on any part of the solution, or do you have other questions?

Here are some related questions to explore:

  1. How do you derive the common ratio from a geometric sequence if you only know two terms?
  2. What happens if the common ratio is negative or a fraction?
  3. How would the formula change if we were looking for the 10th term instead?
  4. Can you prove why the formula for the nn-th term of a geometric sequence works?
  5. How does the sum of the first nn terms of a geometric sequence differ from the nn-th term?

Tip: The common ratio in a geometric sequence determines the growth or decay rate, and if it's greater than 1, the terms increase; if less than 1, the terms decrease.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometric Sequences
Exponents
Algebra

Formulas

a_n = a_1 * r^(n-1)

Theorems

Geometric Sequence Formula

Suitable Grade Level

Grades 7-10