Sequences：练习题

The first four terms of the sequence of triangular numbers are 1, 3, 6, and 10. Find the missing term in the sequence:
1, 3, 6, 10, __, 21, 28

Find an expression for the \(n\)th term of the sequence:
2, 10, 30, 68, ...

Find an expression for the \(n\)th term of the quadratic sequence:
5, 12, 23, 38, 57, ...

The \(n\)th term of a sequence is given by the formula \(T_n = 100 - n^2\).
Calculate the 7th term of this sequence.

A sequence is formed using a rule similar to the Fibonacci sequence: each term is the sum of the two previous terms.
Given the sequence: 1, 4, 5, 9, 14, ...
Find the next two terms.

The \(n\)th term of a sequence is given by the formula \(T_n = n^3 + 2\).
Find the value of the 5th term of this sequence.

The first term of a sequence is 5. Each subsequent term is found by multiplying the previous term by 2 and then subtracting 3. Calculate the 6th term of this sequence.

Find an expression for the \(n\)th term of the following quadratic sequence: \(1, 6, 13, 22, 33, \dots\)

Consider the sequence of square numbers 1, 4, 9, 16, ...
(a) Write down the 12th term.
(b) The \(k\)th term of this sequence is 625. Find the value of \(k\).
(c) Find the \(n\)th term of a new sequence 3, 6, 11, 18, ... by comparing it to the square numbers.
(d) Explain why 150 cannot be a term in the sequence of square numbers.

Find the \(n\)th term for the sequence 2, 9, 28, 65, ...