Sequences：练习题

What is the 50th term of the sequence defined by \( T(n) = n^2 - n \)?

A sequence is defined by the recurrence relation \( a_{n+1} = a_n + 3n - 1 \) for \( n \ge 1 \). If the first term \( a_1 = 5 \), find the general term \( a_n \) of the sequence.

A sequence is given by \( 3, 9, 27, 81, \dots \). What is the general term \( T(n) \) of the sequence?

The first four terms of a sequence are \( 2, 5, 8, 11, \dots \). Which of the following is the general term \( T(n) \) of the sequence?

Find the \( 20^{th} \) term of the sequence defined by the general term \( T(n) = 3n - 5 \).

A sequence is given by the pattern: \( 5, 11, 17, 23, \dots \). Write down the general term \( T(n) \) of this sequence.

A sequence of numbers is given by the formula \( a_n = n^2 - 3n \). Find the value of the 10th term in the sequence.

Find the 5th term of the sequence defined by the general term \( a_n = 2n^2 - 3 \).

The general term of a sequence is given by \( T(n) = 5n - 2 \).

(a) Write down the first three terms of the sequence.
(b) Determine whether 148 is a term in this sequence. If it is, find the value of \( n \).

A pattern of shapes is made from sticks. The number of sticks used for the \( n \)-th shape is given by a linear sequence. The 2nd shape uses 7 sticks and the 5th shape uses 16 sticks.
(a) Find the common difference and the first term of the sequence.
(b) Find the formula for the \( n \)-th term.
(c) How many sticks are used in the 100th shape?