Take n = 4. X1 = 1, x2 = 2, x3 =3 x4 = 4. Lets verify if it actualyy 0 or nor

Let's verify the equation for \( n = 4 \) with \( x_1 = 1 \), \( x_2 = 2 \), \( x_3 = 3 \), and \( x_4 = 4 \).

The equation is:

\[
\left(\frac{3}{4}\right)^{n-1}x_1n^2 + \frac{n^2}{4}\sum_{i=2}^n \left(\frac{3}{4}\right)^{n-i} x_i + 2\sum_{i=1}^n (ix_i) - (2n + 1)\sum_{i=1}^n x_i = 0
\]

### Step 1: Compute each term separately

#### Term 1: \(\left(\frac{3}{4}\right)^{n-1}x_1n^2\)

For \( n = 4 \) and \( x_1 = 1 \):

\[
\left(\frac{3}{4}\right)^{4-1} \times 1 \times 4^2 = \left(\frac{3}{4}\right)^3 \times 16
\]

Calculate:

\[
\left(\frac{3}{4}\right)^3 = \frac{27}{64}
\]

Thus:

\[
\frac{27}{64} \times 16 = \frac{27 \times 16}{64} = \frac{432}{64} = 6.75
\]

#### Term 2: \(\frac{n^2}{4}\sum_{i=2}^n \left(\frac{3}{4}\right)^{n-i} x_i\)

For \( n = 4 \):

\[
\frac{4^2}{4} \sum_{i=2}^4 \left(\frac{3}{4}\right)^{4-i} x_i = 4 \sum_{i=2}^4 \left(\frac{3}{4}\right)^{4-i} x_i
\]

The sum:

\[
4 \left[ \left(\frac{3}{4}\right)^{2} \times 2 + \left(\frac{3}{4}\right)^{1} \times 3 + \left(\frac{3}{4}\right)^{0} \times 4 \right]
\]

Calculate each term:

- \( \left(\frac{3}{4}\right)^2 = \frac{9}{16} \), so \( \frac{9}{16} \times 2 = \frac{18}{16} = 1.125 \)
- \( \left(\frac{3}{4}\right) = 0.75 \), so \( 0.75 \times 3 = 2.25 \)
- \( \left(\frac{3}{4}\right)^0 = 1 \), so \( 1 \times 4 = 4 \)

Thus:

\[
4 \times (1.125 + 2.25 + 4) = 4 \times 7.375 = 29.5
\]

#### Term 3: \(2\sum_{i=1}^n (ix_i)\)

For \( n = 4 \):

\[
2 \sum_{i=1}^4 (i \times x_i) = 2 \times (1 \times 1 + 2 \times 2 + 3 \times 3 + 4 \times 4)
\]

Calculate:

\[
2 \times (1 + 4 + 9 + 16) = 2 \times 30 = 60
\]

#### Term 4: \(-(2n + 1)\sum_{i=1}^n x_i\)

For \( n = 4 \):

\[
-(2 \times 4 + 1) \sum_{i=1}^4 x_i = -9 \times (1 + 2 + 3 + 4)
\]

Calculate:

\[
-9 \times 10 = -90
\]

### Step 2: Combine the results

Combine all the terms:

\[
6.75 + 29.5 + 60 - 90 = 96.25 - 90 = 6.25
\]

### Conclusion:

The calculated result is **not** zero, but instead \( 6.25 \). This indicates that the expression does **not** equal zero for \( n = 4 \) and the given values of \( x_1, x_2, x_3, x_4 \).

---

Do you want details or have any questions?

### Related Questions:
1. What could be the reason that the expression doesn't equal zero for this particular example?
2. How does the choice of \( x_i \) affect the outcome of the equation?
3. Can the equation be verified for other values of \( n \)?
4. Is there a way to generalize the condition under which the expression equals zero?
5. Could there be an error in the initial problem statement?

### Tip:
Always double-check calculations for complex expressions, as even small errors can lead to incorrect conclusions.

Verify the equation with n = 4, x1 = 1, x2 = 2, x3 = 3, and x4 = 4. Detailed step-by-step solution including computations and conclusions. Explore why the expression does not equal zero.

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