Finding n for given sum value

4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\).

1. Find the value of \(S _ { 30 }\) correct to 3 decimal places.
2. Find the least value of \(N\) for which \(S _ { N } > 4.9\).

4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\). Find

1. the value of \(S _ { 30 }\) correct to 3 decimal places,
2. the least value of \(N\) for which \(S _ { N } > 4.9\).

10 Using an algebraic method, determine the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } \geqslant 0.49\).

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1. Express \(\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 }\) as a single fraction.
2. Find how many terms of the series $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 3 \times 5 } + \frac { 2 } { 5 \times 7 } + \ldots + \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) } + \ldots$$ are needed for the sum to exceed 0.999999.

4. The positive integer \(N\) is such that \(1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + N ^ { 2 } = ( 3 N - 2 ) ^ { 2 }\). Write down and simplify an equation satisfied by \(N\). Hence find the possible values of \(N\).

5. Given that $$\sum _ { r = k } ^ { 76 } ( r - 31 ) = 980$$ show that there are two possible values of \(k\).
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1. Given that $$\frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { A } { 2 r - 1 } + \frac { B } { 2 r + 1 }$$ find the values of \(A\) and \(B\).
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
3. Find the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) differs from 0.5 by less than 0.001 .

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1. Use the method of differences to show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { ( r - 1 ) r ( r + 1 ) } = \frac { 1 } { 4 } - \frac { 1 } { 2 n } + \frac { 1 } { 2 ( n + 1 ) }$$ [5 marks]
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2. Find the smallest integer \(n\) such that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { ( r - 1 ) r ( r + 1 ) } > 0.24999$$