Solve n(n+1)(2n+1)/6

Question

Simplify the expression

\frac{2 n ^{3} + 3 n ^{2} + n}{6}

Evaluate

n (n + 1) \times \frac{2 n + 1}{6}

Multiply the terms

\frac{n ( 2 n + 1 )}{6} (n + 1)

Multiply the terms

\frac{n ( 2 n + 1 ) ( n + 1 )}{6}

Solution

More Steps

Evaluate

n (2 n + 1) (n + 1)

Multiply the terms

More Steps

Evaluate

n (2 n + 1)

Apply the distributive property

n \times 2 n + n \times 1

Multiply the terms

2 n^{2} + n \times 1

Any expression multiplied by 1 remains the same

2 n^{2} + n

(2 n^{2} + n) (n + 1)

Apply the distributive property

2 n^{2} \times n + 2 n^{2} \times 1 + n \times n + n \times 1

Multiply the terms

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Evaluate

n^{2} \times n

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

n^{2 + 1}

Add the numbers

n^{3}

2 n^{3} + 2 n^{2} \times 1 + n \times n + n \times 1

Any expression multiplied by 1 remains the same

2 n^{3} + 2 n^{2} + n \times n + n \times 1

Multiply the terms

2 n^{3} + 2 n^{2} + n^{2} + n \times 1

Any expression multiplied by 1 remains the same

2 n^{3} + 2 n^{2} + n^{2} + n

Add the terms

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Evaluate

2 n^{2} + n^{2}

Collect like terms by calculating the sum or difference of their coefficients

(2 + 1) n^{2}

Add the numbers

3 n^{2}

2 n^{3} + 3 n^{2} + n

\frac{2 n ^{3} + 3 n ^{2} + n}{6}

Show Solution

Find the roots

n_{1} = - 1, n_{2} = - \frac{1}{2}, n_{3} = 0

Alternative Form

n_{1} = - 1, n_{2} = - 0.5, n_{3} = 0

Evaluate

n (n + 1) \times \frac{2 n + 1}{6}

To find the roots of the expression,set the expression equal to 0

n (n + 1) \times \frac{2 n + 1}{6} = 0

Multiply the terms

More Steps

Multiply the terms

n (n + 1) \times \frac{2 n + 1}{6}

Multiply the terms

\frac{n ( 2 n + 1 )}{6} (n + 1)

Multiply the terms

\frac{n ( 2 n + 1 ) ( n + 1 )}{6}

\frac{n ( 2 n + 1 ) ( n + 1 )}{6} = 0

Simplify

n (2 n + 1) (n + 1) = 0

Separate the equation into 3 possible cases

n = 0 2 n + 1 = 0 n + 1 = 0

Solve the equation

More Steps

Evaluate

2 n + 1 = 0

Move the constant to the right-hand side and change its sign

2 n = 0 - 1

Removing 0 doesn't change the value,so remove it from the expression

2 n = - 1

Divide both sides

\frac{2 n}{2} = \frac{- 1}{2}

Divide the numbers

n = \frac{- 1}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

n = - \frac{1}{2}

n = 0 n = - \frac{1}{2} n + 1 = 0

Solve the equation

More Steps

Evaluate

n + 1 = 0

Move the constant to the right-hand side and change its sign

n = 0 - 1

Removing 0 doesn't change the value,so remove it from the expression

n = - 1

n = 0 n = - \frac{1}{2} n = - 1

Solution

n_{1} = - 1, n_{2} = - \frac{1}{2}, n_{3} = 0

Alternative Form

n_{1} = - 1, n_{2} = - 0.5, n_{3} = 0

Show Solution