If the 8th term of an AP is 36 and the 16th term is 68.
Find:
a) the first term
b) the common difference
c) The 20th term
(b) clearly d=4, since
T16-T8 = 8d = 32
Now do (a) and (c)
For (c), note that T20-T16 = 4d
To find the first term and the common difference of an arithmetic progression (AP), we can use the formulas:
a) The nth term of an AP is given by the formula: an = a + (n-1)d, where "a" is the first term and "d" is the common difference.
b) The difference between any two terms in an AP is the common difference: d = an - a(n-1)
Given the information in the problem, we can solve for the first term and the common difference.
a) To find the first term (a), we can use the formula an = a + (n-1)d with n = 8 and an = 36:
36 = a + (8-1)d
36 = a + 7d
b) To find the common difference (d), we can use the formula d = an - a(n-1) with n = 8, an = 36, and n = 16, an = 68:
d = 68 - a(16-1)
d = 68 - 15a
c) To find the 20th term, we can use the formula an = a + (n-1)d with n = 20:
an = a + 19d
Now, let's solve for a and d using the given information.
a) Substitute the values in the equation 36 = a + 7d:
36 = a + 7d
b) Substitute the values in the equation d = 68 - 15a:
d = 68 - 15a
To find the values of a and d, we need another equation. We can substitute the value of d in terms of a from the second equation into the first equation.
36 = a + 7(68 - 15a)
36 = a + 476 - 105a
36 + 105a = 476
105a = 476 - 36
105a = 440
a = 440 / 105
a = 4.19 (approximately)
Now, substitute the value of a in the equation d = 68 - 15a to find the value of d:
d = 68 - 15(4.19)
d = 68 - 62.85
d = 5.15 (approximately)
So, the first term (a) is approximately 4.19 and the common difference (d) is approximately 5.15.
c) To find the 20th term (a20), we can substitute the values of a and d in the equation an = a + (n-1)d:
a20 = 4.19 + (20-1)(5.15)
a20 = 4.19 + 19(5.15)
a20 = 4.19 + 97.85
a20 = 102.04
So, the 20th term (a20) is approximately 102.04.
To find the first term, common difference, and the 20th term of an arithmetic progression (AP) given the 8th and 16th terms, you can use the formulas for the nth term and the arithmetic mean.
a) First, let's find the first term (a):
The nth term formula for an arithmetic progression is given by:
An = a + (n - 1)d
We are given that the 8th term (A8) is 36. Plugging the values into the formula, we have:
36 = a + (8 - 1)d
Simplifying the equation, we get:
36 = a + 7d
b) Next, let's find the common difference (d):
We are also given that the 16th term (A16) is 68. Plugging the values into the formula, we have:
68 = a + (16 - 1)d
Simplifying the equation, we get:
68 = a + 15d
c) Finally, let's find the 20th term (A20):
We can now use the values of a and d that we have calculated to find the 20th term. Plugging the values into the nth term formula, we have:
A20 = a + (20 - 1)d
Now, using the values of a and d from the equations above, we can calculate A20.
Here's a breakdown of the steps to solve for a, d, and A20:
Step 1: Solve the equation a + 7d = 36 for a. This will give us the value of the first term (a).
Step 2: Solve the equation a + 15d = 68 for d. This will give us the common difference (d).
Step 3: Plug the values of a and d into the equation a + 19d to find the 20th term (A20).
By following these steps, you will be able to find the answers to all three parts of the question.