the second term of AP is 15 and the fifth term is 21.find
(1) the common difference and the first term
(2) The eight term
(3) The sum of the first 16 term
To find the common difference and the first term, we can use the formula for the nth term of an arithmetic progression (AP):
nth term = first term + (n - 1) * common difference
Let's solve the given problem step by step:
(1) Finding the common difference and the first term:
Given information:
Second term (n = 2) = 15
Fifth term (n = 5) = 21
We can set up the formulas for the second and fifth terms and solve for the common difference (d) and first term (a):
For the second term:
15 = a + (2 - 1) * d ---> Equation 1
For the fifth term:
21 = a + (5 - 1) * d ---> Equation 2
Simplify Equation 1:
15 = a + d
Simplify Equation 2:
21 = a + 4d
We now have a system of two equations. We can solve this system using the method of substitution or elimination.
Start by subtracting Equation 1 from Equation 2 to eliminate the "a" term:
21 - 15 = (a + 4d) - (a + d)
6 = 3d
Divide both sides by 3:
d = 2
Substitute the value of d = 2 into Equation 1 to solve for the first term:
15 = a + (2 - 1) * 2
15 = a + 2
Subtract 2 from both sides:
a = 13
Therefore, the common difference (d) is 2 and the first term (a) is 13.
(2) Finding the eighth term:
We can use the same formula for the nth term of the AP: nth term = a + (n - 1) * d
Substituting the known values:
n = 8
a = 13
d = 2
eighth term = 13 + (8 - 1) * 2
eighth term = 13 + (7 * 2)
eighth term = 13 + 14
eighth term = 27
Therefore, the eighth term of the AP is 27.
(3) Finding the sum of the first 16 terms:
To find the sum of the first n terms of an AP, we can use the formula:
Sum(n) = (n / 2) * [2a + (n - 1) * d]
Substituting the known values:
n = 16 (number of terms)
a = 13 (first term)
d = 2 (common difference)
Sum(16) = (16 / 2) * [2 * 13 + (16 - 1) * 2]
Sum(16) = 8 * [26 + 15 * 2]
Sum(16) = 8 * [26 + 30]
Sum(16) = 8 * 56
Sum(16) = 448
Therefore, the sum of the first 16 terms of the AP is 448.
T2 and T5 differ by 3d = 6
Now you can easily get d and a, and
S16 = 8(2a+15d)