Question

Here are the first five terms of an arithmetic sequence.

2, 7, 12, 17, 22

The nth term of a different arithmetic sequence is 4n+ 15

The last term of each sequence is
the same number.

There are the same number of terms in each sequence.

Find the number of terms in each sequence.

Answer 1

18 terms in both sequences

Explanation:

first sequence is given by 2 + (n-1) 5 = -3 +5n

because 2 is first number and 5 is the common difference.

let's find first few terms of 4n +15 to see what's going on:

4(1) + 15 = 19, 4(2) + 15 = 23, 4(3) + 15 = 27......

these have a common difference of 4.

so for each nth term in both sequences, the first sequence will 'close the gap' to the second sequence by 1 every time.

first term in 1st sequence = 2

first term in 2nd sequence = 19

the gap is 19 -2 = 17

closing the gap by one for 17th term gives:

1st sequence = -3 + 5(17) = 82

2nd sequence = 4(17) + 15 = 85

for 18th term:

1st sequence = -3 + 5(18) = 87

2nd sequence = 4(18) + 15 = 87

now they are equal.

that means that there are 18 terms in both sequences