Sequences：練習題

Consider the following sequence of numbers:
2, 5, 8, 11, ...

Find the next term in this sequence.

A sequence of shapes is made using small squares. Shape 1 uses 3 squares, Shape 2 uses 8 squares, and Shape 3 uses 15 squares. If this pattern continues, find an expression for the number of squares in Shape \(n\).

Find the \(n^{th}\) term of the sequence: 0, 4, 20, 54, 112, ...

A sequence is defined by the following terms:
5, 11, 19, 29, 41, ...

By finding the first and second differences, determine the expression for the \(n^{th}\) term of this quadratic sequence.

Consider the sequence: 0, 10, 28, 54, 88, ...

By finding the first and second differences, find an expression for the \(n^{th}\) term of this sequence.

Write down the next two terms in the following sequence:
2, 5, 8, 11, ...

Find the \( n^{\text{th}} \) term expression for the following linear sequence:
\( -2, 1, 4, 7, \dots \)

Consider the sequence whose first four terms are:
1, 14, 51, 124, ...

Find an expression for the \(n^{th}\) term of this sequence.

A pattern of squares is made using sticks as follows:

Pattern 1: 1 square (4 sticks)
Pattern 2: 2 squares in a row (7 sticks)
Pattern 3: 3 squares in a row (10 sticks)

(a) Find the number of sticks needed for Pattern 5.
(b) Determine the n^th term rule for the number of sticks, \(S_n\), in Pattern \(n\).
(c) Calculate the number of sticks needed for Pattern 50.
(d) A pattern uses 121 sticks. Find the number of squares in this pattern.

The first four terms of an exponential sequence are 5, 15, 45, 135, ...

(a) Find the common ratio.
(b) Find an expression for the \(n^{\text{th}}\) term, \(T_n\).
(c) Find the smallest value of \(n\) for which the term is greater than 10,000.