<ol class="abc-list"><li><div class="list-item-paragraph">The sum, ${{S}_{n}}$, of the first $n$ terms of an arithmetic progression is given by ${{S}_{n}}=pn+q{{n}^{2}}$.</div><div class="list-item-paragraph">Given also that ${{S}_{3}}=6$ and ${{S}_{5}}=11$,</div><ol class="i-list"><li><div class="list-item-paragraph">Find the values of $p$ and $q$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce, or find otherwise, an expression for the $n$-th term and the value of the common difference.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">Find the set of values of $\theta $ lying in the interval $-\frac{1}{2}\pi <\theta <\frac{1}{2}\pi $ such that the sum to infinity of the geometric series $1+\sin \theta +{{\sin }^{2}}\theta +\ldots $ is greater than $2$.</div><p style="text-align: right;">[4]</p></li></ol>

The sum, ${{S}_{n}}$, of the first $n$ terms of an arithmetic progression is given by ${{S}_{n}}=pn+q{{n}^{2}}$.

Given also that ${{S}_{3}}=6$ and ${{S}_{5}}=11$,

Find the set of values of $\theta $ lying in the interval $-\frac{1}{2}\pi <\theta <\frac{1}{2}\pi $ such that the sum to infinity of the geometric series $1+\sin \theta +{{\sin }^{2}}\theta +\ldots $ is greater than $2$.

N1984 P1 Q1

N1984 P1 Q1

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