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Given the arithmetic progression (AP): [tex]4, x, 0[/tex]

1. Find the value of [tex]x[/tex] in the AP.

2. Find the [tex]7^{th}[/tex] term of the AP.

3. If the last term is 20, find the number of terms in the AP.

4. Write the formula for the nth term of the AP.

Answer :

Sure, let's solve the problem step by step!

We're dealing with an arithmetic progression (AP) with a sequence given as [tex]\(4, x, 0\)[/tex].

### Step 1: Find the value of [tex]\(x\)[/tex].

1. Determine the common difference [tex]\(d\)[/tex]:
- The sequence is given as [tex]\(4, x, 0\)[/tex].
- In an AP, the difference between consecutive terms is constant (this is the common difference).
- Using the third term, which is 0, and the first term, which is 4, we can set up the equation for the third term:
[tex]\[
0 = 4 + 2d
\][/tex]
- Solve for [tex]\(d\)[/tex]:
[tex]\[
2d = 0 - 4 \quad \Rightarrow \quad 2d = -4 \quad \Rightarrow \quad d = -2
\][/tex]

2. Calculate [tex]\(x\)[/tex] using the common difference:
- The second term [tex]\(x\)[/tex] is calculated using the first term and the common difference:
[tex]\[
x = 4 + d
\][/tex]
- Substitute [tex]\(d = -2\)[/tex] into the equation:
[tex]\[
x = 4 - 2 = 2
\][/tex]

### Step 2: Formula for the [tex]\(k\)[/tex]-th term of the AP.

The formula for the [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) of an arithmetic progression with first term [tex]\(a_1\)[/tex] and common difference [tex]\(d\)[/tex] is:
[tex]\[
a_k = a_1 + (k-1) \cdot d
\][/tex]
For our sequence, [tex]\(a_1 = 4\)[/tex] and [tex]\(d = -2\)[/tex].

### Step 3: Find the number of terms if the last term is 20.

We are given the last term as -20. We use the formula for the [tex]\(n\)[/tex]-th term and set it equal to -20:
[tex]\[
-20 = 4 + (n-1) \cdot (-2)
\][/tex]

1. Rearrange the equation to solve for [tex]\(n\)[/tex]:
[tex]\[
-20 = 4 - 2(n-1)
\][/tex]
[tex]\[
-20 = 4 - 2n + 2
\][/tex]
[tex]\[
-20 = 6 - 2n
\][/tex]
[tex]\[
-20 - 6 = -2n
\][/tex]
[tex]\[
-26 = -2n
\][/tex]
[tex]\[
n = \frac{-26}{-2} = 13
\][/tex]

Therefore, there are 13 terms in the arithmetic progression where the last term is -20.

This provides a detailed computation for all parts of the question.

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Rewritten by : Barada