The sum of the first 5 terms of an arthmetic sequence, is 30 and the sum of the next 4 terms is 69 .
Find the arthemetic progression?
so the sum of the first 9 terms would be 99
sum(5) = 30 ---> (5/2)(2a + 4d) = 30
5(2a+4d) = 60
2a + 4d = 12
a + 2d = 6
sum(9) = 99 ----> (9/2)(2a + 8d) = 99
2a + 8d = 22
a + 4d = 11
subtract them
2d = 5
d = 5/2 or 2.5
then a = 1
the arithmetic sequence is
1 , 3.5, 6, 8.5 , ...
What is lhe sum of lhe first 12 terms of an A.p whose first term is 15 and common difference is 13.
1038
To find the arithmetic progression, we need to determine the common difference (d) and the first term (a1).
Let's start by finding the sum of the first 5 terms:
Given: the sum of the first 5 terms is 30.
The sum of an arithmetic sequence can be calculated using the formula:
Sn = (n/2) * (2a1 + (n-1)d)
Substituting the values:
30 = (5/2) * (2a1 + (5-1)d)
60 = 2a1 + 4d
2a1 + 4d = 60 -- Equation (1)
Next, let's find the sum of the next 4 terms:
Given: the sum of the next 4 terms is 69.
Using the same formula:
Sn = (n/2) * (2a1 + (n-1)d)
Substituting the values:
69 = (4/2) * (2a5 + (4-1)d)
69 = 2a5 + 3d
2a5 + 3d = 69 -- Equation (2)
Now, we have two equations with two unknowns (a1 and d). Let's solve them simultaneously.
From Equation (1), we have:
2a1 = 60 - 4d
a1 = 30 - 2d -- Equation (3)
Substituting Equation (3) into Equation (2), we have:
2(30 - 2d) + 3d = 69
60 - 4d + 3d = 69
-d = 9
d = -9
Substituting the value of d into Equation (3):
a1 = 30 - 2(-9)
a1 = 30 + 18
a1 = 48
Therefore, the first term (a1) of the arithmetic progression is 48, and the common difference (d) is -9.