Is −21 a term of the arithmetic sequence having t1=90, t3=81?
every odd-numbered term is a multiple of 9, so no.
you can do the math, noting that d = -9/2, so you want n such that
90 - 9/2 n = -21
You will find that n is not an integer
To determine if -21 is a term of the arithmetic sequence, we need to find the common difference.
Given:
t₁ = 90
t₃ = 81
The general formula to find a term of an arithmetic sequence is:
tₙ = t₁ + (n - 1) * d
where:
tₙ is the nth term of the sequence
t₁ is the first term of the sequence
n is the position of the desired term
d is the common difference
Let's find the common difference (d) first using the given information:
t₃ = t₁ + (3 - 1) * d
81 = 90 + 2d
Now, solve for d:
2d = 81 - 90
2d = -9
d = -9/2
The common difference is -9/2 or -4.5.
Now, substitute d into the formula for the nth term to check if -21 is a term:
-21 = 90 + (n - 1) * (-4.5)
Simplify the equation:
-21 = 90 - 4.5n + 4.5
-21 = 94.5 - 4.5n
Rearrange the equation:
4.5n = 94.5 + 21
4.5n = 115.5
n = 115.5 / 4.5
n = 25.67
Since n must be a whole number (the position of the term), -21 is not a term of the arithmetic sequence with t₁ = 90 and t₃ = 81.
To determine if −21 is a term of the arithmetic sequence, we'll first need to identify the common difference (d) of the sequence. Once we know the common difference, we can use it to calculate the terms of the sequence.
The formula to find the nth term of an arithmetic sequence is given by:
tn = a + (n - 1)d,
where tn is the term we want to calculate, a is the first term of the sequence, n is the position of the term in the sequence, and d is the common difference of the sequence.
Given that:
t1 = 90, and
t3 = 81,
we will find the common difference (d) as follows:
t1 = a + (1 - 1)d = a,
t3 = a + (3 - 1)d = a + 2d.
From the provided information, we can set up two equations:
90 = a,
81 = a + 2d.
Now, let's solve these equations to find the common difference (d):
From the first equation, we get: a = 90.
Substituting the value of a in the second equation, we have:
81 = 90 + 2d.
Rearranging this equation gives us:
2d = 81 - 90,
2d = -9,
d = -9/2.
Now that we know the common difference (d = -9/2), we can find the values of the terms in the sequence. Let's check if −21 is one of them:
Substituting the values of a, n, and d in the formula, we have:
tn = 90 + (n - 1)(-9/2).
Setting tn equal to −21, we get:
−21 = 90 + (n - 1)(-9/2).
Now, let's solve this equation for n:
−21 = 90 - (9/2)n + 9/2.
Multiplying both sides of the equation by 2 to remove the fraction, we get:
-42 = 180 - 9n + 9.
Combining like terms, we have:
-42 = 189 - 9n.
Adding 9n to both sides and rearranging the terms, we get:
9n = 189 + 42,
9n = 231.
Dividing both sides by 9, we find:
n = 231/9,
n = 25.67.
Since n is not a whole number, −21 is not a term in the arithmetic sequence with t1 = 90 and t3 = 81.
Therefore, the answer to the question is no, −21 is not a term of the arithmetic sequence.