8.6 Patterns

Find the next two terms of the sequence 3, 9, 15, 21, … and describe the rule.

The $n$-th term of a sequence is $T_n = 3n + 2$. Find $T_12$.

A pattern of dots has 4 dots in the 1st, 7 dots in the 2nd, 10 dots in the 3rd. (a) How many dots in the 4th? (b) Describe the rule.

Find the next term of: 2, 4, 8, …

Continue: $-3, -1, 1, 3, \ldots$

For the sequence with formula $T_n = 2n + 5$, find the first three terms.

A pattern of stars: 1st has 5, 2nd has 8, 3rd has 11. Find a rule.

Find the common difference of the sequence 12, 8, 4, 0, −4, …

Verify whether 47 is a term of the sequence $T_n = 3n + 2$.

Verify whether 40 is a term of the sequence $T_n = 3n + 2$.

Find a formula for the $n$-th term of the sequence 5, 9, 13, 17, …

Find the 20th term of $T_n = 5n - 2$.

A pattern of matchsticks: each new triangle adds 2 sticks. The first uses 3 sticks. Find a formula for $T_n$.

A sequence has first term 10 and common difference $-3$. Write a formula for $T_n$.

Find $n$ when $T_n = 71$ in the sequence $T_n = 4n - 5$.

A linear sequence has $T_3 = 11$ and $T_7 = 27$. (a) Find $d$. (b) Find $T_1$. (c) Write $T_n$.

A square pattern grows: 1st has 1 dot, 2nd has 4, 3rd has 9. Write a formula.

A sequence: $1, 3, 6, 10, 15, \ldots$ (triangular numbers). Find $T_6$ and $T_7$.

A sequence of stair-step shapes: 1st uses 1 cube, 2nd uses 3, 3rd uses 6, 4th uses 10. Find the rule.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, … State the recursion and find $T_8$.

A row of conference tables seats people. The pattern is $M = 6n + 2$. (a) How many people sit at 6 tables? (b) How many tables are needed to seat at least 80 people?

In Mr. Packer's arrangement, $a$ people sit on each side, $b$ at each end. Show $P = 2an + 2b$ and find $P$ when $a = 4$, $b = 2$, $n = 6$.

Patterns of matchsticks: 1st 3 sticks, 2nd 5, 3rd 7. (a) Formula for $T_n$. (b) Which pattern uses 99 sticks?

Find a formula and the 50th term of: 7, 11, 15, 19, …

A growing pattern of squares uses 4 sticks for the first square, then adds 3 sticks per extra square in a row. (a) Formula for $T_n$. (b) How many squares can you make from 100 sticks? (c) How many sticks left over?

Find a formula for $T_n$ in the sequence $-1, 2, 5, 8, 11, \ldots$

The 4th term of a linear sequence is 17 and the 10th term is 41. Find a formula for $T_n$.

The sequence $5, 8, 13, 20, 29, \ldots$ has second differences 2. Find a formula for $T_n$.

Find the sum of the first 10 terms of $T_n = 2n + 1$ (odd numbers starting from 3).

For the pattern $T_n = n^2 - n$, find $T_1, T_2, T_5, T_{10}$.

The sequence of perfect squares is $1, 4, 9, 16, 25, \ldots$ (a) Formula for $T_n$. (b) Differences between consecutive terms. (c) Use this to find $T_{20}$ without squaring.

The 5th term of a linear sequence is 18 and the 12th term is 53. Find a formula, and find the smallest $n$ for which $T_n > 100$.

For the sequence $1, 5, 12, 22, 35, \ldots$ (pentagonal numbers, formula $P_n = \tfrac{n(3n-1)}{2}$): verify the formula for $n = 1, 2, 3$, and find $P_{10}$.

Find the sum of all multiples of 3 between 1 and 100 (inclusive).

A linear sequence has first term $a$ and common difference $d$. (a) Express $T_5$ in terms of $a$ and $d$. (b) If $T_5 = 23$ and $T_{12} = 65$, find $a$ and $d$.

Two arithmetic sequences A: 5, 8, 11, … and B: 2, 7, 12, … . (a) Find formulas. (b) For which $n$ are the $n$-th terms equal?

A figurate-number pattern: the $n$-th hexagonal number is $H_n = n(2n-1)$. Find $H_6$ and $H_{10}$.

A geometric sequence has first term 3 and common ratio 2. (a) Find the first 5 terms. (b) Find a formula $T_n$. (c) Find $T_{10}$.

A staircase pattern uses 1, 4, 9, 16, … cubes per layer. (a) State the formula for the $n$-th layer. (b) Find the total number of cubes in the first 5 layers. (c) Find a formula for the total $S_n = 1^2 + 2^2 + \ldots + n^2$ if you can.

A sequence has $T_n = 2n^2 - n$. (a) Find $T_1, T_2, T_3$. (b) Show that the second differences are constant and find their value. (c) Is 119 a term of the sequence?