If O is the origin anf `A_(n)` is the point with coordinates `(n,n+1)` then `(OA_(1))^(2)+(OA_(2))^(2)+…..+(OA_(7))^(2)` is equal to

Answer

To solve the problem, we need to calculate the sum of the squares of the distances from the origin \( O(0, 0) \) to the points \( A_n(n, n+1) \) for \( n = 1 \) to \( 7 \).

### Step-by-Step Solution:

1. **Identify the Coordinates**:
The coordinates of the points \( A_n \) are given by \( A_n(n, n+1) \). Thus, we have:
- \( A_1(1, 2) \)
- \( A_2(2, 3) \)
- \( A_3(3, 4) \)
- \( A_4(4, 5) \)
- \( A_5(5, 6) \)
- \( A_6(6, 7) \)
- \( A_7(7, 8) \)

2. **Calculate the Distance Squared**:
The distance from the origin \( O(0, 0) \) to a point \( A_n(x, y) \) is given by the formula:
\[ OA_n^2 = x^2 + y^2 \]
For each \( A_n \):
- \( OA_1^2 = 1^2 + 2^2 = 1 + 4 = 5 \)
- \( OA_2^2 = 2^2 + 3^2 = 4 + 9 = 13 \)
- \( OA_3^2 = 3^2 + 4^2 = 9 + 16 = 25 \)
- \( OA_4^2 = 4^2 + 5^2 = 16 + 25 = 41 \)
- \( OA_5^2 = 5^2 + 6^2 = 25 + 36 = 61 \)
- \( OA_6^2 = 6^2 + 7^2 = 36 + 49 = 85 \)
- \( OA_7^2 = 7^2 + 8^2 = 49 + 64 = 113 \)

3. **Sum the Distances Squared**:
Now, we need to sum these values:
\[ OA_1^2 + OA_2^2 + OA_3^2 + OA_4^2 + OA_5^2 + OA_6^2 + OA_7^2 = 5 + 13 + 25 + 41 + 61 + 85 + 113 \]

4. **Calculate the Total**:
Performing the addition step-by-step:
- \( 5 + 13 = 18 \)
- \( 18 + 25 = 43 \)
- \( 43 + 41 = 84 \)
- \( 84 + 61 = 145 \)
- \( 145 + 85 = 230 \)
- \( 230 + 113 = 343 \)

5. **Final Result**:
Therefore, the value of \( (OA_1)^2 + (OA_2)^2 + \ldots + (OA_7)^2 \) is \( 343 \).

### Summary:
The final answer is:
\[ \boxed{343} \]