Final answer:
The third term of the arithmetic progression (AP) is 2.
Step-by-step explanation:
To find the third term of an arithmetic progression (AP) with a given sum and sum of squares, we need to use the formulas for the sum of an AP and the sum of squares of an AP.
Let the third term be 'a', and let the common difference be 'd'
From the given information, we have:
a + (a+d) + (a+2d) + (a+3d) + (a+4d) = 35
a^2 + (a+d)^2 + (a+2d)^2 + (a+3d)^2 + (a+4d)^2 = 285
Simplifying these equations, we get:
5a + 10d = 35
5a^2 + 30ad + 30d^2 = 285
Using the first equation, we can express 'a' in terms of 'd' as:
a = (35 - 10d)/5 = 7 - 2d
Substituting this value of 'a' in the second equation, we can solve for 'd':
5(7 - 2d)^2 + 30(7 - 2d)d + 30d^2 = 285
Expanding this equation, we have:
140 - 70d + 10d^2 + 210d - 60d^2 + 30d^2 = 285
Combining like terms, we get:
-20d^2 + 140d - 145 = 0
Solving this quadratic equation, we find:
d = 5/2 or d = - 7/2
Since 'd' represents the common difference, it cannot be negative. Therefore, 'd' must be 5/2.
Substituting this value of 'd' back into the equation for 'a', we get:
a = 7 - 2(5/2) = 7 - 5 = 2
Therefore, the third term of the arithmetic progression is 'a' which is equal to 2.
Therefore answer is b) 7.