prove that the squares of all terms of the arithmetic sequence 4,7,10.....belong to the sequence

To prove that the squares of all terms of the arithmetic sequence \(4, 7, 10, \dots\) also form an arithmetic sequence, we need to do the following:

### Step 1: Define the general term of the sequence
The given sequence is an arithmetic sequence with the first term \(a_1 = 4\) and a common difference \(d = 3\). The general term \(a_n\) of this sequence can be expressed as:
\[
a_n = a_1 + (n-1) \times d = 4 + (n-1) \times 3 = 3n + 1
\]

### Step 2: Square the general term
We need to find the square of each term in this sequence:
\[
a_n^2 = (3n + 1)^2
\]
Expanding the square:
\[
a_n^2 = (3n + 1)^2 = 9n^2 + 6n + 1
\]

### Step 3: Determine if the squared terms form an arithmetic sequence
For the squares of the terms \(a_n^2\) to form an arithmetic sequence, the difference between consecutive squared terms should be constant.

Let's calculate the difference between \(a_{n+1}^2\) and \(a_n^2\):
\[
a_{n+1} = 3(n+1) + 1 = 3n + 4
\]
\[
a_{n+1}^2 = (3n + 4)^2 = 9n^2 + 24n + 16
\]
Now, subtract \(a_n^2\) from \(a_{n+1}^2\):
\[
a_{n+1}^2 - a_n^2 = (9n^2 + 24n + 16) - (9n^2 + 6n + 1)
\]
Simplifying this:
\[
a_{n+1}^2 - a_n^2 = 18n + 15
\]
This expression shows that the difference between the squares of consecutive terms is \(18n + 15\), which is not constant but depends on \(n\). 

### Conclusion:
Since the difference between consecutive squared terms is not constant, the squares of the terms of the original arithmetic sequence do **not** themselves form an arithmetic sequence. Thus, the statement that the squares of all terms of the arithmetic sequence belong to the sequence is incorrect.

---

Would you like more details or have any questions? Here are 5 related questions you might find interesting:

1. What is the definition of an arithmetic sequence?
2. How do we prove that a sequence is arithmetic?
3. What other sequences, besides arithmetic, exist?
4. Can the sum of the squares of an arithmetic sequence form an arithmetic sequence?
5. How do we determine if a given sequence is geometric?

**Tip:** Always verify if the difference between consecutive terms in a sequence is constant when proving that the sequence is arithmetic.

Prove that the squares of all terms of the arithmetic sequence 4, 7, 10... belong to the sequence.

Explore the proof that the squares of terms from the arithmetic sequence 4, 7, 10... do not form another arithmetic sequence. Understand key algebraic concepts like arithmetic sequences, quadratic expressions, and how to determine if a sequence remains arithmetic. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation