The fifth term of an arithmetic progression is three times the second term,and the third term is 10.a)What is the first term,b)the common difference and c)the 15th term?

a) 12

b) 2
c) 214

a) 2

b) 4
c ) 58

To find the first term, common difference, and the 15th term of an arithmetic progression, we need to use the given information.

Let's assume that the first term of the arithmetic progression is represented by 'a', and the common difference is represented by 'd'.

a) To find the first term, we need to use the information that the third term is 10. In an arithmetic progression, the nth term can be found using the formula: an = a + (n - 1)d, where 'a' is the first term, 'n' is the term number, and 'd' is the common difference.

Given that the third term (n = 3) is 10, we can substitute these values into the formula:

a3 = a + (3 - 1)d
10 = a + 2d

Now, we have one equation with two variables. However, there's additional information that states that the fifth term is three times the second term. Using the same formula, we can find the fifth term:

a5 = a + (5 - 1)d

But since we know that a5 is three times the second term (a2), we can write:

a5 = 3a2

Now, we have two equations with two variables:

10 = a + 2d ...(1)
a + 4d = 3a ...(2)

By solving these equations simultaneously, we can find the values of 'a' and 'd'.

b) To calculate the common difference, we need to solve the equations we obtained in step 'a'. Subtracting equation (2) from equation (1), we get:

10 - (a + 4d) = a + 2d - 3a

Simplifying the equation, we have:

10 - a - 4d = -2a + 2d

Combining like terms:

d = -(3/6)a

So, the common difference is -(3/6)a, which simplifies to -(1/2)a.

c) To find the 15th term (a15) of the arithmetic progression, we will use the formula mentioned earlier:

a15 = a + (15 - 1)d
= a + 14d

To find the exact value of a15, we need to substitute the value of 'd' calculated in step 'b'.

a15 = a + 14(-1/2)a
= a - 7a
= -6a

Therefore, the 15th term is -6a.

In summary:
a) The first term is 'a'.
b) The common difference is -(1/2)a.
c) The 15th term is -6a.