Given thatT7=23, Tn=43 and T2n=91. Find a,d and n
To find the values for a, d, and n, we need to use the formulas for the nth term of an arithmetic sequence.
The formula for the nth term of an arithmetic sequence is:
Tn = a + (n - 1)d
Given:
T7 = 23
Tn = 43
T2n = 91
Step 1: Finding the common difference (d)
We are given T7 = 23, so we can substitute into the formula to find the value of T7:
23 = a + (7 - 1)d
Simplifying, we have:
23 = a + 6d
Step 2: Finding n
We are given Tn = 43, so we can substitute into the formula to find the value of n:
43 = a + (n - 1)d
Step 3: Finding the value of T2n
We are given T2n = 91, so we can substitute into the formula to find the value of T2n:
91 = a + (2n - 1)d
Step 4: Solving the system of linear equations
Now, we have three equations with three unknowns (a, d, and n). We can solve this system of equations to find the values of a, d, and n.
From equations (1) and (2), we have:
a + 6d = 23 (equation 1)
a + (n - 1)d = 43 (equation 2)
From equations (2) and (3), we have:
a + (n - 1)d = 43 (equation 2)
a + (2n - 1)d = 91 (equation 3)
Now we can solve these equations simultaneously.
By subtracting equation (2) from equation (1), we have:
(a + 6d) - (a + (n - 1)d) = 23 - 43
6d - (n - 1)d = -20
6d - nd + d = -20
(d - nd) + 7d = -20
(7 - n)d = -20
By subtracting equation (2) from equation (3), we have:
(a + (2n - 1)d) - (a + (n - 1)d) = 91 - 43
(2n - 1)d - (n - 1)d = 48
(2nd - d + d = 48
2nd = 48
Now we can solve these two equations for d:
(7 - n)d = -20 (equation 4)
2nd = 48 (equation 5)
From equation (4):
7d - nd = -20
From equation (5):
2nd = 48
Simplifying equation (5):
nd = 24
By applying substitution, we can express d in terms of n:
7d - (nd) = -20
7d - 24 = -20
7d = 4
d = 4/7
Now we can find n by substituting d into equation (5):
(4/7)n = 24
n = 24 * (7/4)
n = 42
Finally, substitute n = 42 into equation (4) to find d:
7d - (42d) = -20
-35d = -20
d = 20/35
d = 4/7
Therefore, the values for a, d, and n are:
a = 23 - 6d = 23 - 6(4/7) = 23 - 24/7 = (161 - 24)/7 = 137/7
d = 4/7
n = 42
plug and chug
a + 6d = 23
a + (n-1)d = 43
a + (2n-1)d = 91
Now just solve for a,d,n