The sixth term of an appointment is 37 and the sum of the first sixth term is147 find the first term,find the sum of the first fifteen terms.
Surely you mean "arithmetic progression" not "appointment"!
a+5d = 37
6/2 (2a+5d) = 147
You want
15/2 (2a+14d)
so solve for a and d and plug 'em in.
a + 5d = 37 ... Eqn 1
6a + 15d = 147 ... Eqn 2
Solve eqn. 1 and 2 simultaneously
a = 12
d = 5
First term = 12
S(15) = 15/2[2a + 14d]
S(15) = 15/2[(2x12) + (14x5)]
S(15) = 705
sum of the first fifteen terms is 705
To solve this problem, we will use two equations based on the information given. Let's denote the first term as 'a' and the common difference between the terms as 'd'.
The formula for finding the nth term of an arithmetic sequence is given by:
an = a + (n-1)d
1) Finding the first term:
Given that the sixth term is 37, we can use the formula:
a6 = a + (6-1)d = 37
Substituting the values, we get:
a + 5d = 37 ----(Equation 1)
2) Finding the sum of the first six terms:
The formula for the sum of the first n terms of an arithmetic sequence is given by:
Sn = (n/2)(2a + (n-1)d)
Given that the sum of the first six terms is 147, we can use the formula and substitute the values:
147 = (6/2)(2a + (6-1)d)
147 = 3(2a + 5d)
147 = 6a + 15d ----(Equation 2)
Now, we have a system of equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of 'a' and 'd'.
Solving the system of equations:
From Equation 1, we can express 'a' in terms of 'd':
a = 37 - 5d
Substituting this value of 'a' into Equation 2:
147 = 6(37 - 5d) + 15d
147 = 222 - 30d + 15d
147 = 222 - 15d
-75 = -15d
d = 5
Now that we have the value of 'd', we can substitute it back into Equation 1 to find the value of 'a':
a + 5(5) = 37
a + 25 = 37
a = 12
So, the first term (a) is 12.
3) Finding the sum of the first fifteen terms:
We can use the formula for the sum of the first n terms of an arithmetic sequence again, but this time with n = 15:
Sn = (15/2)(2a + (15-1)d)
Substituting the values into the formula:
S15 = (15/2)(2(12) + (15-1)(5))
S15 = (15/2)(24 + 14(5))
S15 = (15/2)(24 + 70)
S15 = (15/2)(94)
S15 = 705
Therefore, the sum of the first fifteen terms is 705.