Math Problem Statement

The function machine below produces a sequence. What is the 20th term in this sequence?

Solution

The image shows a function machine that produces a sequence, where the position in the sequence is first multiplied by 4, and then 3 is added. This is a linear transformation, and we can write it as:

Term=(Position×4)+3\text{Term} = (\text{Position} \times 4) + 3

To find the 20th term, we substitute the position value (20) into this equation:

Term20=(20×4)+3=80+3=83\text{Term}_{20} = (20 \times 4) + 3 = 80 + 3 = 83

So, the 20th term in the sequence is 83.

Would you like more details on how this works or have any further questions? Here are some related questions for deeper understanding:

  1. How would the sequence change if the multiplier was different, like 5 instead of 4?
  2. What if we added a larger number, say 10, instead of 3? How would that affect the terms?
  3. How would you find the nth term in a sequence like this, where the operation is a linear transformation?
  4. What is the general form for any arithmetic sequence that uses multiplication and addition?
  5. How can we graph this sequence, and what would its graph look like?

Tip: A quick way to solve for terms in a sequence like this is to use the formula derived from the function machine directly for any position, rather than calculating each term step by step.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Sequences
Linear Transformations
Arithmetic Sequences

Formulas

Term = (Position × 4) + 3

Theorems

Linear Transformation Theorem

Suitable Grade Level

Grades 5-7