the thirteenth term of an arythmeticprogression is 27. given that the seventh term equals to three times the second term, find:
the first term nd the common difference of the progression
a+12d = 27
a+6d = 3(a+d)
Now just solve for a and d.
To find the first term and common difference of the arithmetic progression, we need to use the given information.
Let's denote the first term as 'a' and the common difference as 'd'.
Given:
- The thirteenth term is 27, so we have: a + 12d = 27. (1)
- The seventh term is three times the second term, so we have: a + 6d = 3(a + d). (2)
Now we can solve these two equations to find the values of 'a' and 'd'.
Step 1: Solve equation (2) for 'a':
a + 6d = 3a + 3d Simplify the equation.
6d - 3d = 3a - a
3d = 2a
Step 2: Substitute the value of 'a' from equation (1) into equation (2):
a + 6d = 3(a + d) Substitute 2a for 3d (from step 1).
a + 6d = 3a + 3d Use distributive property.
6d - 3d = 3a - a Simplify the equation.
3d = 2a
Step 3: Solve equation 6d - 3d = 3a - a for 'a':
3d = 2a
3d/2 = a
So, we have found the value of 'a' in terms of 'd'.
Step 4: Substitute the value of 'a' in equation (1) with 3d/2:
a + 12d = 27 Substitute 3d/2 for 'a'.
(3d/2) + 12d = 27 Multiply through by the denominator 2 to remove the fraction.
3d + 24d = 54
27d = 54
d = 54/27
d = 2
Therefore, the common difference, 'd', is 2.
Step 5: Substitute the value of 'd' in equation (2) to find 'a':
a + 6(2) = 3(a + 2)
a + 12 = 3a + 6
12 - 6 = 3a - a
6 = 2a
a = 6/2
a = 3
Therefore, the first term, 'a', is 3.
Hence, the first term of the arithmetic progression is 3 and the common difference is 2.