Find the General term tn if t5=10 and t20=5
To find the general term tn, we can use the formula for the nth term of an arithmetic sequence:
tn = a + (n-1)d
Here, a represents the first term, and d represents the common difference between consecutive terms.
Given t5 = 10:
Using the formula, we can substitute a = t1, n = 5, and tn = 10:
10 = t1 + (5-1)d
Similarly, given t20 = 5:
Substituting a = t1, n = 20, and tn = 5:
5 = t1 + (20-1)d
We now have two equations with two unknowns: t1 and d. We can solve these equations simultaneously to find their values.
First, we subtract the first equation from the second equation:
5 - 10 = (t1 + (20-1)d) - (t1 + (5-1)d)
-5 = (t1 + 19d) - (t1 + 4d)
Simplifying further:
-5 = t1 + 19d - t1 - 4d
-5 = 15d
Dividing both sides by 15:
d = -5/15
d = -1/3
Now, we can substitute the value of d back into any of the original equations to solve for t1:
10 = t1 + (5-1)d
10 = t1 + 4(-1/3)
10 = t1 - 4/3
Multiplying both sides by 3:
30 = 3t1 - 4
Adding 4 to both sides:
34 = 3t1
Dividing both sides by 3:
t1 = 34/3
Therefore, the general term tn is given by:
tn = (34/3) + (n-1)(-1/3)
Simplifying further:
tn = (34 - n + 1)/3
So, the general term tn is (34 - n + 1)/3.