If the sum of 3th and 8th terms of an A.p is 7 and the sum of 7th and 14th term is -3,find the 10th term .
term3 + term8 = 7
a+2d + a+7d = 7
2a + 9d = 7 **
term7 + term14 = -3
a+6d + a+13d = -3
2a + 19d = -3 ***
subtract ** from ***
10d =-10
d = -1
sub into *** to get a , then find a + 9d
To find the 10th term of an arithmetic progression (A.P.), we need to determine the common difference first.
Let's assume that the first term of the A.P. is 'a', and the common difference is 'd'.
The 3rd term can be written as: a + 2d
The 8th term can be written as: a + 7d
Given that the sum of the 3rd and 8th terms is 7, we can form the equation:
(a + 2d) + (a + 7d) = 7
2a + 9d = 7 ---(Equation 1)
Similarly, the 7th term can be written as: a + 6d
The 14th term can be written as: a + 13d
Given that the sum of the 7th and 14th terms is -3, we can form the equation:
(a + 6d) + (a + 13d) = -3
2a + 19d = -3 ---(Equation 2)
Now we have a system of two linear equations in two variables (a and d). We can solve this system of equations to find the values of 'a' and 'd'.
Multiplying Equation 1 by 19 and Equation 2 by 9, we get:
38a + 171d = 133 ---(Equation 3)
18a + 171d = -27 ---(Equation 4)
Subtracting Equation 4 from Equation 3, we eliminate 'd' and get:
(38a - 18a) + (171d - 171d) = 133 - (-27)
20a = 160
a = 8
Substituting the value of 'a' (8) in Equation 1, we can solve for 'd':
2a + 9d = 7
2(8) + 9d = 7
16 + 9d = 7
9d = 7 - 16
9d = -9
d = -1
Now we know the values of 'a' (first term) and 'd' (common difference) in the arithmetic progression:
First term (a) = 8
Common difference (d) = -1
To find the 10th term, we use the formula for the nth term of an A.P.:
tn = a + (n - 1)d
Substituting the values:
t10 = 8 + (10 - 1)(-1)
t10 = 8 + 9(-1)
t10 = 8 - 9
t10 = -1
Therefore, the 10th term of the arithmetic progression is -1.