Sequences and series：練習問題

Using the formula for the sum of the first \( n \) natural numbers, calculate the value of \( \sum_{r=1}^{40} r \).

Find the sum of the first 12 terms of a geometric series where the first term is 5 and the common ratio is 2.

The sum of the first \(n\) terms of a series is given by \(S_n = 3^n - 1\). Show that the series is geometric and find the common ratio \(r\).

A sequence is defined by the recurrence relation \( u_{n+1} = 2u_n - 3 \) with \( u_1 = 4 \). Calculate the value of \( u_4 \).

A sequence is defined by the recurrence relation \(u_{n+1} = k u_n + 3\), where \(k\) is a constant. Given that \(u_1 = 4\) and \(u_3 = 13\), find the two possible values of \(k\).

A geometric sequence has first term \( a = 4 \) and common ratio \( r = 0.5 \). Find the exact value of the sum to infinity of this series.

A convergent geometric series has a second term of \( 4 \) and a sum to infinity of \( 18 \). Find the two possible values of the common ratio \( r \).

Determine the first three terms in the binomial expansion of \( \frac{1+x}{(1-2x)^4} \) in ascending powers of \( x \) for \( |x| < \frac{1}{2} \).

A geometric series has first term \( a \) and common ratio \( r \). The sum of the second and third terms of the series is 15. The sum to infinity of the series is 64.
(a) Show that \( 64r^2 - 49r + 15 = 0 \).
(b) Find the two possible values of \( r \).
(c) For each value of \( r \), find the corresponding first term \( a \).

A geometric sequence is such that the sum of its first three terms is 35 and the product of its first three terms is 1000.
(a) Find the two possible values for the common ratio, \( r \).
(b) For the case where the sequence is convergent, find the sum to infinity.