Find first term and common difference of arithmetic sequence whose 13th them is 159 and 20th is 264.
a + 12 d = 159
a + 19 d = 264
---------------
7 d = 105
d = 15
a = 159-12(15) = -21
To find the first term and common difference of an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
An = A1 + (n - 1) * d
Where An is the nth term, A1 is the first term, n is the position of the term in the sequence, and d is the common difference.
Given that the 13th term is 159 and the 20th term is 264, we can substitute these values into the formula to get two equations:
159 = A1 + (13 - 1) * d (equation 1)
264 = A1 + (20 - 1) * d (equation 2)
We now have a system of two equations with two unknowns (A1 and d). We can solve this system to find the values.
First, let's simplify equation 1:
159 = A1 + 12d (equation 1)
Now let's simplify equation 2:
264 = A1 + 19d (equation 2)
We can use these simplified equations to solve for A1 and d simultaneously. Subtracting equation 1 from equation 2 will eliminate A1:
264 - 159 = (A1 + 19d) - (A1 + 12d)
105 = 7d
Divide both sides of the equation by 7:
15 = d
Now that we have found the value of d, we can substitute it back into equation 1 to solve for A1:
159 = A1 + 12 * 15
159 = A1 + 180
Subtract 180 from both sides of the equation:
-21 = A1
Therefore, the first term (A1) of the arithmetic sequence is -21, and the common difference (d) is 15.