Math Problem Statement

  1. Find the specified term of the following

a) Identify the 7th term of the sequence: 4, 20, 100, ...

Solution

The given sequence is:

4,20,100,4, 20, 100, \dots

We notice that each term is multiplying by 5, so this could be a geometric sequence. Let’s verify that.

To confirm:

  • 204=5\frac{20}{4} = 5
  • 10020=5\frac{100}{20} = 5

Hence, this is indeed a geometric sequence with the first term a=4a = 4 and the common ratio r=5r = 5.

The general formula for the nn-th term of a geometric sequence is:

Tn=arn1T_n = a \cdot r^{n-1}

Where:

  • TnT_n is the nn-th term,
  • aa is the first term,
  • rr is the common ratio, and
  • nn is the position of the term.

Now, to find the 7th term:

T7=4571=456T_7 = 4 \cdot 5^{7-1} = 4 \cdot 5^6

Let’s compute 565^6:

56=156255^6 = 15625

Now calculate the 7th term:

T7=415625=62500T_7 = 4 \cdot 15625 = 62500

Thus, the 7th term is:

T7=62500T_7 = 62500

Would you like more details or have any questions?

Here are 5 related questions to explore:

  1. What is the 10th term of this geometric sequence?
  2. Can you derive the formula for the sum of the first nn terms of a geometric sequence?
  3. How does changing the common ratio affect the growth of a geometric sequence?
  4. What happens if the common ratio is a fraction between 0 and 1?
  5. How do geometric sequences compare to arithmetic sequences in terms of growth?

Tip: Always check for the common ratio or difference when identifying a sequence type—geometric or arithmetic.

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Math Problem Analysis

Mathematical Concepts

Geometric Sequences
Exponents

Formulas

T_n = a * r^(n-1)
Exponential Growth Formula

Theorems

Geometric Sequence Term Formula

Suitable Grade Level

Grades 9-11