Find the sum of the following series 1 square + 4 square + 7 square + 10 square to n teram?

Question

Answer

Sum of Series of Squares

To find the sum of the series of squares 1², 4², 7², 10², and so on up to n terms, we can use the formula for the sum of the squares of the first n natural numbers:

Formula: 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6

Using the Formula:

To use the formula, we need to identify the last term of the series, which is n². We can express this in terms of the first term (1²) and the common difference (which is 3), as follows:

n² = (1² + (n-1)3)²

Expanding this expression and simplifying, we get:

n² = 9n² - 6n + 1
or
8n² - 6n + 1 = 0

Solving for n using the quadratic formula, we get:

n = [6 ± √(36 - 32)]/16
n = [6 ± √4]/16
n = (3/4) or (1/2)

Since n must be a positive integer, we take n = 3.

Final Calculation:

Now, we can use the formula to find the sum of the series:

1² + 4² + 7² + ... + (3n-2)² = n(n+1)(2n+1)/6

Substituting n = 3, we get:

1² + 4² + 7² + 10² + 13² + 16² = 3(4)(7)/6
1² + 4² + 7² + 10² + 13² + 16² = 84

Therefore, the sum of the series is 84.

Answer

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