the sixth term of arithmetical progression is half of its twelfth term.the first term is equal to
To find the first term of an arithmetic progression given the sixth term and the twelfth term, we need to use the formula for the nth term of an arithmetic progression.
Let's denote the first term as 'a', the common difference as 'd', the sixth term as 'A6', and the twelfth term as 'A12'.
The general formula for the nth term of an arithmetic progression is:
An = a + (n - 1) * d
We are given that the sixth term (A6) is half of the twelfth term (A12). Mathematically, we can represent this as:
A6 = 1/2 * A12
Substituting the values into the general formula, we get:
a + (6 - 1) * d = 1/2 * (a + (12 - 1) * d)
Simplifying further:
a + 5d = 1/2 * (a + 11d)
To solve for 'a', let's eliminate 'd' by subtracting the left-hand side (LHS) from both sides of the equation:
5d - 11d = 1/2 * a - a
-6d = -1/2 * a
Now, let's simplify the equation by dividing both sides by -6:
d = (1/2 * a) / 6
d = (1/2) * (a/6)
d = a/12
We now have the expression for 'd' in terms of 'a'.
Since 'a' refers to the first term, an arithmetic progression can have infinitely many solutions depending on the value of 'a'.
In conclusion, the first term ('a') can be any real number.