Question

Arithmetic sequences (nth term)​

Answer 1

Tick the arithmetic sequences.

In order for a sequence to be arithmetic the relations between their numbers must be the sum of a common ratio, therefore to determine which ones are arithmetic sequence we need to satisfy the following expression:

$a_n - a_{n-1} = a_{n-2} - a_{n-3}$

The expression just mean that the subtraction of consecutive terms should be equal for position in the sequence.

"1, 5, 9, 13...":

$13 - 9 = 5 - 1\\4 = 4$

Since the expression is valid, then this is a arithmetic sequence.

"6, 10, 15, 21...:

$21 - 15 = 10 - 6\\6 = 4$

Since the expression is invalid, then this isn't a arithmetic sequence.

"2, 3, 5, 8...:

$8 - 5 = 3 - 2\\3 = 1$

Since the expression is invalid, then this isn't a arithmetic sequence.

"2, -4, 8, -16...:

$-16- 8 = -4 - 2\\-24 = -6$

Since the expression is invalid, then this isn't a arithmetic sequence.

"-1, 2, 5, 8...:

$8 - 5 = 2 -(- 1)\\3 = 3$

Since the expression is valid, then this is a arithmetic sequence.

"73, 66, 59, 52...:

$52 - 59 = 66 - 73\\-7 = -7$

Since the expression is valid, then this is a arithmetic sequence.

"6, 1, -4, -9...:

$-9 - (-4) = 1 - 6\\-5 = -5$

Since the expression is valid, then this is a arithmetic sequence.

"1, 2, 4, 8...:

$8 - 4 = 2 - 1\\4 = 1$

Since the expression is invalid, then this isn't a arithmetic sequence.

Find the first five terms of the patterns with these nth terms.

For $3n + 3$:

$3*1 + 3 = 3 + 3 = 6$

$6 + 3 = 9$

$9 + 3 = 12$

$12 + 3 = 15$

$15 + 3 = 18$

(6, 9, 12, 15, 18)

For $9n - 7$:

$9*1 - 7 = 9 - 7 = 2$

$2 + 9 = 11$

$11 + 9 = 20$

$20 + 9 = 29$

$29 + 9 = 38$

(2,11,20,29,38)

Write down a formula for the nth term of these patterns. The first term is n = 1.

The nth term of any arithmetic sequence is: $a(n) = a(1)*(n - 1)*r$

Therefore we need to identify r in each sequence.

For (9,15,21,27,33):

$r = 15 - 9 = 6$

$a(n) = 1*(n - 1)*6 \\a(n) = 6*n - 6$

For (-2,-8,-14,-20,-26):

$r = -8 - (-2) = -6$

$a(n) = 1*(n - 1)*(-6)\\a(n) = -6*n + 6$