| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Standard +0.3 This is a straightforward application of standard techniques: partial fractions decomposition of 1/(n(n+1)) yields the telescoping form 2(1/n - 1/(n+1)), then method of differences gives immediate cancellation. While it requires two techniques in sequence, both are routine A-level procedures with no conceptual obstacles or novel insight required. |
| Spec | 4.05c Partial fractions: extended to quadratic denominators4.06b Method of differences: telescoping series |
1 The $n$th triangular number, $T _ { n }$, is given by the formula $T _ { n } = \frac { 1 } { 2 } n ( n + 1 )$.\\
(i) Express $\frac { 1 } { T _ { n } }$ in terms of partial fractions.\\
(ii) Hence, using the method of differences, show that $\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q1}}