Series (Edexcel IAL Further Maths): Exam Questions

Use the standard results for $\sum_{r = 1}^{n} r$ and for $\sum_{r = 1}^{n} r^{3}$ to show that, for all positive integers $n$,

$\sum_{r = 1}^{n} r \left(r^{2} - 3\right) = \frac{n}{4} \left(n + a\right) \left(n + b\right) \left(n + c\right)$

where $a$, $b$ and $c$ are integers to be found.

In this question use the standard results for summations.

Show that for all positive integers equation$n$

$\sum_{r = 1}^{n} ( 12 r^{2} + 2 r - 3 ) = A n^{3} + B n^{2}$

where $A$and $B$ are integers to be determined.

Hence determine the value of $n$ for which

$\sum_{r = 1}^{2 n} r^{3} - \sum_{r = 1}^{n} ( 12 r^{2} + 2 r - 3 ) = 270$

Use the standard results for summations to show that, for all positive integers $n$,

$\sum_{r = 1}^{n} r ( 2 r^{2} - 3 r - 1 ) = 1 \frac{1}{2} n ( n + 1 )^{2 } ( n - 2 )$

Hence show that, for all positive integers $n$,

$\sum_{r = n}^{2 n} r ( 2 r^{2} - 3 r - 1 ) = 1 \frac{1}{2} n ( n - 1 ) ( a n + b ) ( c n + d )$

where $a$, $b$, $c$ and $d$ are integers to be determined.

Use the standard results for $\sum_{r = 1}^{n} r^{2}$ and $\sum_{r = 1}^{n} r^{3}$ to show that, for all positive integers $n$

$\sum_{r = 1}^{n} r^{2} ( r + 2 ) = \frac{1}{12} n ( n + 1 ) ( a n^{2} + b n + c )$

where $a$, $b$ and $c$ are integers to be determined.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use the standard results for $\sum_{r = 1}^{n} r$ and $\sum_{r = 1}^{n} r^{2}$ to show that for all positive integers equation$n$

$\sum_{r = 1}^{n} \left(\right. 7 r - 5 \left.\right)^{2} = \frac{n}{6} \left(7 n + 1\right) \left(A n + B\right)$

where $A$and $B$ are integers to be determined.

Use the standard results for $\sum_{r = 1}^{n} r^{2}$ and $\sum_{r = 1}^{n} r$ to show that for all positive integers $n$

$\sum_{r = 0}^{n} \left(r + 1\right) \left(r + 2\right) = \frac{1}{3} \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)$

Hence determine the value of

$10\times11+11\times12+12\times13+\dots+100\times101$

Prove by induction that, for $n\inℕ$

$\sum_{r = 1}^{n} r^{3} = 1 \frac{1}{4} n^{2} ( n + 1 )^{2}$

Using the standard summation formulae, show that

$\sum_{r = 1}^{n} r ( r + 1 ) ( r - 1 ) = 1 \frac{1}{4} n ( n + A ) ( n + B ) ( n + C )$

where $A$, $B$ and $C$ are constants to be determined.

Determine the value of $n$ for which

$3 \sum_{r = 1}^{n} r ( r + 1 ) ( r - 1 ) = 17 \sum_{r = n}^{2 n} r^{2}$

Use the standard results for SUM_ r = 1 to n of r ^3 , SUM_ r = 1 to n of r ^2 text and end text SUM_ r = 1 to n of r to show that for all positive integers $n$,

$\sum_{r = 1}^{n} r ( r - 1 ) ( r - 3 ) = \frac{1}{12} n ( n + 1 ) ( n - 1 ) ( 3 n - 10 )$

Hence show that

$\sum_{r = n + 1}^{2 n + 1} r ( r - 1 ) ( r - 3 ) = \frac{1}{12} n ( n + 1 ) ( a n^{2} + b n + c )$

where $a$, $b$ and $c$ are integers to be determined.

Prove by induction that for $n\inℕ$

$\sum_{r = 1}^{n} r^{2} = \frac{n}{6} ( n + 1 ) ( 2 n + 1 )$

Hence show that

$\sum_{r = 1}^{n} ( r^{2} + 2 ) = \frac{n}{6} ( a n^{2} + b n + c )$

where $a$, $b$ and $c$ are integers to be found.

Using your answers to part (b), find the value of

$\sum_{r = 10}^{25} ( r^{2} + 2 )$

Using the formulae for SUM_ r = 1 to n of r text and end text SUM_ r = 1 to n of r ^2, show that

$\sum_{r = 1}^{n} ( r + 1 ) ( r + 5 ) = \frac{n}{6} ( n + 7 ) ( 2 n + 7 )$

for all positive integers $n$.

Hence show that

$\sum_{r = n + 1}^{2 n} ( r + 1 ) ( r + 5 ) = \frac{7 n}{6} ( n + 1 ) ( a n + b )$

where $a$ and $b$ are integers to be determined.

Use the standard results for $\sum_{r = 1}^{n} r^{2}$ and $\sum_{r = 1}^{n} r$ to show that

$\sum_{r = 1}^{n} \left(\right. 2 r - 1 \left.\right)^{2} = \frac{1}{3} n \left(4 n^{2} - 1\right)$

for all positive integers $n$.

Hence find the exact value of the sum of the squares of the odd numbers between 200 and 500.