An arithmetic sequence has a first term of 0 and a 10th term of 15. If 10 is an output of the sequence, which term number is it? Show and explain how you know you have the correct answer.
the general term of an AS is
a + (n-1)d, where a is the first term, d is the common difference, and n is the number of terms
so :
a = 0
a+9d = 15
0+9d = 15
d = 15/9 = 5/3
For 10 to be one of the terms
a + (n-1)d = 10
0 + (n-1)(5/3) = 10
times 3:
5(n-1) = 30
5n-5 = 30
5n = 35
n = 7
check:
term(7) = a+6d
= 0 + 6(5/3) = 10
All is good!
To find the term number for which the output is 10 in the given arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
nth term = a + (n - 1)d
where a is the first term, n is the term number, and d is the common difference.
Given that the first term (a) is 0, we can substitute these values into the formula:
10 = 0 + (n - 1)d
Simplifying the equation, we have:
10 = (n - 1)d
Now, we need to find the common difference (d). We know that the 10th term is 15, which can be expressed as:
15 = 0 + (10 - 1)d
Simplifying this equation, we have:
15 = 9d
Solving for d, we get:
d = 15 / 9
d ≈ 1.667 (rounded to 3 decimal places)
Now, substituting the value of d back into the previous equation, we have:
10 = (n - 1)(1.667)
Simplifying this equation, we get:
10 = 1.667n - 1.667
Adding 1.667 to both sides, we have:
11.667 = 1.667n
Dividing both sides by 1.667, we get:
n ≈ 6.996 (rounded to 3 decimal places)
Since term numbers must be whole numbers, we can conclude that 10 is not a term of this arithmetic sequence.
Therefore, there is no correct answer for this question as there is no term number for which the output is 10 in this given arithmetic sequence.