determine the arithmetic sequence that has the seventh term equal to 3 and the twelfth term equal to -3
a+6d = 3
a+11d = -3
clearly, 5d = -6, so d = -6/5
that means a = 51/5
To determine the arithmetic sequence, we need to find the common difference (d) between its terms.
The formula to find the nth term (Tn) of an arithmetic sequence is:
Tn = a + (n - 1) * d
Where a represents the first term and n represents the term number.
We are given that the seventh term (T7) is 3 and the twelfth term (T12) is -3.
Using the formula, we can create two equations:
T7 = a + (7 - 1) * d ... (equation 1)
T12 = a + (12 - 1) * d ... (equation 2)
Substituting the given values:
3 = a + 6d ... (equation 3)
-3 = a + 11d ... (equation 4)
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the method of elimination:
Multiply equation 3 by 11, and equation 4 by 6 to eliminate the variable 'a':
33 = 11a + 66d ... (equation 5)
-18 = 6a + 66d ... (equation 6)
Now, subtract equation 6 from equation 5 to eliminate the variable 'd':
33 - (-18) = 11a - 6a
51 = 5a
Divide both sides of the equation by 5:
a = 10.2
Now, substitute the value of 'a' back into one of the original equations, equation 3, for example:
3 = 10.2 + 6d
Subtract 10.2 from both sides:
- 7.2 = 6d
Divide both sides by 6:
-1.2 = d
So, the first term (a) of the arithmetic sequence is 10.2 and the common difference (d) is -1.2.
The arithmetic sequence is:
10.2, 9, 7.8, 6.6, 5.4, 4.2, 3, 1.8, 0.6, -0.6, -1.8, -3