Introduction to Sequence Calculation

Understanding the Sequence Formula

Consider the sequence where pn n^2 - 3n^2 2n 2014. To avoid confusion, let's first simplify this expression:

Step 1: Simplify the Expression

Upon simplification, we obtain:

pn 4n^2 - 2n - 2014

Step 2: Seperate and Identify the Series

Next, we want to calculate the sum of this sequence from p1 to p100. To do this, we need to break down the expression into manageable parts:

pn 2014n - 2n^2 4n^2

Thus, we can write:

pn 4n^2 - 2n - 2014

Step 3: Utilize Known Summation Formulas

We know the following formulas:

Sum of the first n natural numbers: (sum_{i1}^n i frac{n(n 1)}{2}) Sum of the first n squares: (sum_{i1}^n i^2 frac{n(n 1)(2n 1)}{6})

Using these formulas, we can separate and sum the terms:

Step 4: Calculate the Sum

Now, we calculate the sum:

(sum_{i1}^n pn 2014n - 2 sum_{i1}^n i 4 sum_{i1}^n i^2)

This becomes:

( 2014n - 2 frac{n(n 1)}{2} 4 frac{n(n 1)(2n 1)}{6})

Clean it up:

( 2014n - (n(n 1)) frac{2n(n 1)(2n 1)}{3})

( 2014n - n^2 - n frac{2n^3 3n^2 n}{3})

( 2014n - n^2 - n frac{2n^3}{3} n^2 frac{n}{3})

( 2014n - n frac{2n^3}{3} frac{n}{3})

( 2014n - n frac{2n^3 n}{3})

( 2014nn - n frac{2n^3 n}{3})

( 2014nnn - n frac{2n^3 n}{3})

Step 5: Plug in n 100

To calculate the sum for n 100, plug in the value:

( sum_{i1}^{100} pn 2014(100) - (100 - 1) frac{2(100)^3 100}{3})

Simplify this expression:

( 201400 - 99 frac{2000000 100}{3} )

( 201301 frac{2000100}{3})

( 201301 666700)

( 868001)

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Conclusion

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