the third term of an arithmetic sequence is 14 and the ninth term is -1. Find the first four terms of the sequence
ninth -1, third 14
it goes now by 15 in six steps; step 15/6 or 2.5
third through tenth terms
14, 11.5, 9, 6.5, 4, 1.5, -1, -3.5,...
check that
third term is 14 and ninth term is -1
14 - 2.5 = 11.5 > 4th term
11.5 - 2.5 = 9 > 5th term
9 - 2.5 = 6.5 > 6th term
6.5 - 2.5 = 4 > 7th term
4 - 2.5 = 1.5 > 8th term
1.5 - 2.5 = -1 > 9th term
The terms are decreasing by 2.5 > (-2.5)
To go from one term to the next, subtract 2.5.
The common difference is -2.5
d = -2.5
Arithmetic Sequence Formula:
Tn = Tn + d(n-1)
a = 1st term
n = nth term
d = common difference
We don't know the first term yet!
Tn = T1 + d(n-1)
Substitute 3 for n, and -2.5 for d
T3 = T1 + -2.5(3-1)
Now substitute 14 for T3
14 = T1 + -2.5(2)
14 = T1 -5
14 + 5 = T1 -5 +5
19 = T1
T1 = 19
T2 = 16.5
T3 = 14
T4 = 11.5
Now to find a term:
Tn = T1 + d(n-1)
T2 = 19 + -2.5(2-1) T3 = 19 + -2.5(3-1)
T2 = 19 + -2.5(1) T3 = 19 + -2.5(2)
T2 = 19 + -2.5 T3 = 19 + -5
T2 = 16.5 T3 = 14
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You can also use this formula:
Tn = dn + 21.5
T1 = -2.5(1) + 21.5
T1 = 19
T4 = -2.5(4) + 21.5
T4 = 11.5
T9 = -2.5(9) + 21.5
T9 = -1
To find the first four terms of the arithmetic sequence, we will use the formula for the nth term:
An = A1 + (n-1)d
where An is the nth term, A1 is the first term, n is the position of the term, and d is the common difference.
Given:
A3 = 14 and A9 = -1
Using the formula for A3:
A3 = A1 + (3-1)d
14 = A1 + 2d
Using the formula for A9:
A9 = A1 + (9-1)d
-1 = A1 + 8d
Now we have a system of equations:
A1 + 2d = 14
A1 + 8d = -1
To solve this system, we can subtract the first equation from the second equation:
(A1 + 8d) - (A1 + 2d) = -1 - 14
6d = -15
Divide both sides by 6:
d = -15/6
d = -2.5
Now substitute this value of d back into one of the original equations to find A1:
A1 + 2(-2.5) = 14
A1 - 5 = 14
A1 = 14 + 5
A1 = 19
Now we can find the first four terms of the sequence:
A1 = 19
A2 = A1 + d = 19 + (-2.5) = 16.5
A3 = A1 + 2d = 19 + 2(-2.5) = 14
A4 = A1 + 3d = 19 + 3(-2.5) = 10.5
Therefore, the first four terms of the sequence are 19, 16.5, 14, and 10.5.
To find the first four terms of the arithmetic sequence, we need to determine two things: the common difference and the first term.
First, we can use the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n - 1)d
Where:
aₙ is the nth term,
a₁ is the first term,
n is the term number, and
d is the common difference.
Given that the third term (a₃) is 14 and the ninth term (a₉) is -1, we can use these values to set up two equations and solve for the two unknowns, a₁ and d.
1) Using a₃ = 14:
14 = a₁ + 2d (since n = 3, so 3 - 1 = 2)
2) Using a₉ = -1:
-1 = a₁ + 8d (since n = 9, so 9 - 1 = 8)
Now, we can solve these two equations to find the values of a₁ and d.
Subtracting equation (1) from equation (2), we get:
-1 - 14 = (a₁ + 8d) - (a₁ + 2d)
-15 = 6d
Dividing both sides by 6, we get:
d = -15/6
Simplifying, we have:
d = -2.5
Substituting the value of d into either equation (1) or equation (2), we can solve for a₁:
14 = a₁ + 2(-2.5)
14 = a₁ - 5
Adding 5 to both sides, we get:
a₁ = 19
So, the first term (a₁) of the arithmetic sequence is 19, and the common difference (d) is -2.5.
Now, we can find the first four terms of the sequence by substituting the values of a₁ and d into the formula for the nth term:
a₁ = 19
a₂ = a₁ + (2-1)d = 19 + (1)(-2.5) = 19 - 2.5 = 16.5
a₃ = a₁ + (3-1)d = 19 + (2)(-2.5) = 19 - 5 = 14
a₄ = a₁ + (4-1)d = 19 + (3)(-2.5) = 19 - 7.5 = 11.5
Therefore, the first four terms of the arithmetic sequence are:
19, 16.5, 14, 11.5