Sequences Exercises

Answer

The step up from one term to the next term is always 1.

Equivalently, the step down from one term to the previous term is always 1:

This is an arithmetic sequence.

Answer

The step up from one term to the next isn't always the same.

That means this is NOT an arithmetic sequence.

Answer

Look at the difference between successive terms:

3 – 1 = 2

5 – 3 = 2

7 – 5 = 2

9 – 7 = 2

Since we always get 2 as the difference, this is an arithmetic sequence.

Answer

Look at the difference between successive terms:

21 – 15 = 6

26 – 21 = 5

30 – 26 = 4

Since we don't get the same difference every time, this is not an arithmetic sequence.

Answer

Look at the step up from one term to the next.

Since the step is always 10, this IS an arithmetic sequence.

Answer

The step from one term to the next alternates between -2 and + 2.

Since the step isn't always the same, this is NOT an arithmetic sequence.

Answer

Look at the difference between successive terms:

-10 – 5 = -15

15 – (-10) = 25

-20 – 15 = -35

Since the difference between successive terms isn't always the same, this is NOT an arithmetic sequence.

Answer

To get from one term to the next, we subtract 2 (or add -2, if you prefer).

Since the step size is always the same, this IS an arithmetic sequence.

Answer

We start at 10 and step up 7 to get to the next term.

The first four terms are 10, 17, 24, 31.

Answer

We start at 5 and step down 2 each time.

The first four terms are 5, 3, 1, -1.

Answer

We start at -1 and step down 3 each time.

The first four terms are -1, -4, -7, -10.

Answer

It takes 5 steps.

Answer

It takes 74 steps. From a₁ to a₂ is 1 step. From a₁ to a₃ is 2 steps. From a₁ to a₄ is 3 steps.

Continuing this pattern, from a₁ to a₇₅ must be 74 steps.

Answer

Continuing the pattern, from a₁ to a_n takes (n – 1) steps.

Answer

It takes 7 steps to get from a₁ to a₈. The step size is d = 3, so

a₈ = a₁ + 7d

= 20 + 7(3)

= 41.

Answer

It takes 10 steps of size d = -1 to get from a₁ to a₁₁, so

a₁₁ = a₁ + 10d

= 1 + 10(-1)

= -9.

Answer

It takes 19 steps of size 2 to get from a₁ to a₂₀, so

a₂₀ = a₁ + 19d

= 5 + 19(2)

= 43.

Answer

It takes 9 steps to get from a₁ to a₁₀. If each step has size -5, then

a₁₀ = a₁ + 9d

= -3 + 9(-5)

= -48.

Answer

It takes 99 steps to get from a₁ to a₁₀₀. If the step size is d = 5, then

a₁₀₀ = a₁ + 99d

= 7 + 99(5)

= 502.

Answer

The step size is

d = a_{101 –} a₁₀₀ = 2.

To get from a₁ to a₁₀₀ is 99 steps of size 2, so

a₁ + 99d = a₁₀₀

a₁ + 99(2) = 204

a₁ = 6.

We have a₁ = 6 and d = 2.

Answer

The step size is

d = a₁₂ – a₁₁ = 11.

There are 10 steps from a₁ to a₁₁, so

a₁ + 10d = a₁₁

a₁ + 10(11) = 22

a₁ = -88.

We have a₁ = -88 and d = 11.

Answer

The step size is

d = a₂₁ – a₂₀ = -5.

It takes 19 steps to get from a₁ to a₂₀, so

a₁ + 19d = a₂₀

a₁ + 19(-5) = -2

a₁ = 93

We have a₁ = 93 and d = -2.

Answer

It takes 5 steps.

Answer

It takes 1 step to get from a₁₀₀ to a₁₀₁, 2 steps to get from a₁₀₀ to a₁₀₂, and so on. It must take 100 steps to get from a₁₀₀ to a₂₀₀.

Answer

Suppose you have a staircase with n stairs, labeled from 1 to n. If you're standing on the m^th stair, there are (n – m) stairs left before you get to the top.

This translates nicely to sequences. If you're on the m^th term of a sequence, and you want to get to the n^th term, that means you have to take n – m steps.

Answer

We have

a₂₅ – a₂₀ = 20.

There are 5 steps from a₂₀ to a₂₅, so we divide by 5 to get

d = 4.

Then we proceed as before:

a₁ + 19d = a₂₀

a₁ + 19(4) = 13

a₁ = -63.

Once we know we have an arithmetic sequence, if we start at a₁ and take (n – 1) steps of size d, we end up on term a_n.

We can write this idea in symbols as

a₁ + (n – 1)d = a_n.

This formula has three unknowns: a₁, n and d. Like a buy two, get one free deal on cups of key lime pie yogurt, any two pieces of information about an arithmetic sequence tells us the third and final piece of information. So, if we know a₁ and d we can find any term a_n we like. We can also work backwards from d and some term a_n (which gives us n) to find a₁.