求和符號 Sigma：練習題

The sum to infinity of a convergent geometric series is \(80\), and the common ratio is \(r = -\frac{1}{4}\).
Find the first term \(a\) of the series.

An arithmetic series has a first term of \( 5 \) and a common difference of \( 4 \). Find the number of terms, \( n \), such that the sum of the first \( n \) terms, \( S_n \), is equal to \( 630 \).

A geometric series has a common ratio \( r \) and first term \( a \). Given that the sum to infinity is \( 4 \) times the second term, and that all terms are positive, find the value of \( r \).

In an arithmetic series, the first term is \(a = 4\) and the common difference is \(d = 5\).
Find the \(21^{\text{st}}\) term of this series.

The first three terms of a geometric series with positive terms are \(x - 1\), \(x + 2\), and \(3x\).
Find the value of \(x\) and the common ratio \(r\).

A geometric series has first term \(a = 54\) and second term \(u_2 = -18\).
Find the exact sum to infinity, \(S_\infty\), of the series.

The sum of the first \(n\) terms of an arithmetic series is given by \(S_n = 2n^2 - 7n\).
Find the value of the \(10^{\text{th}}\) term, \(u_{10}\).

The sum to infinity of a convergent geometric series is \( 12 \) and the first term is \( 4 \). Determine the range of values for \( n \) such that the sum of the first \( n \) terms, \( S_n \), exceeds \( 11.9 \).

The third term of a geometric series is \(36\) and the common ratio is \(\frac{2}{3}\).
(a) Find the first term of the series.
(b) Find the sum to infinity, \(S_\infty\), of the series.

A geometric series has first term \(a\) and positive common ratio \(r\).
Given that the sum of the first two terms is \(15\) and the sum of the third and fourth terms is \(60\),
(a) find the value of \(r\),
(b) find the value of \(a\),
(c) calculate the least value of \(n\) for which the sum of the first \(n\) terms, \(S_n\), exceeds \(5000\).