Math Problem Statement

Let {ai} with index starting at 0 represent a geometric sequence. If a7 = 7 and a8 = 28, what is a9?

Solution

Given that {ai}\{a_i\} represents a geometric sequence, we know that the ratio between consecutive terms is constant. Let the common ratio be denoted as rr.

The terms provided are:

  • a7=7a_7 = 7
  • a8=28a_8 = 28

In a geometric sequence, we have the relation: an+1=anra_{n+1} = a_n \cdot r Using this, we can express a8a_8 in terms of a7a_7: a8=a7ra_8 = a_7 \cdot r Substituting the given values: 28=7r28 = 7 \cdot r Solving for rr: r=287=4r = \frac{28}{7} = 4

Now, to find a9a_9, we use the fact that: a9=a8ra_9 = a_8 \cdot r Substitute a8=28a_8 = 28 and r=4r = 4: a9=284=112a_9 = 28 \cdot 4 = 112

Thus, a9=112a_9 = 112.

Would you like further details or have any questions?

Here are 5 related questions:

  1. How is the common ratio in a geometric sequence defined?
  2. What is the general formula for the nn-th term of a geometric sequence?
  3. How can you determine if a sequence is geometric from its terms?
  4. What happens to a geometric sequence if the common ratio is less than 1?
  5. How do geometric sequences differ from arithmetic sequences?

Tip: In geometric sequences, each term is found by multiplying the previous term by a constant factor (the common ratio).

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometric Sequences
Ratios

Formulas

a_{n+1} = a_n * r
r = a_{n+1} / a_n

Theorems

Geometric Sequence Definition

Suitable Grade Level

Grades 8-10