The fifth term of an arithmetic progression is three times the first term. Find the sum of the progression given that a quater of the fifth term is 9
a+4d = 3a
(a+4d)/4 = 9
so clearly 3a=36, so a=12, making d=6
APs have no end, so there is no sum.
Let's solve this step-by-step:
Step 1: Let's assume the first term of the arithmetic progression is "a" and the common difference is "d".
Step 2: According to the given information, the fifth term is three times the first term. So, the fifth term is 3a.
Step 3: We are also given that a quarter of the fifth term is 9. So, (1/4)(3a) = 9. Simplifying this equation, we get: 3a/4 = 9.
Step 4: To find the value of "a", we can multiply both sides of the equation by 4/3: (4/3)(3a/4) = (4/3)(9). This simplifies to: a = 12.
Step 5: Now that we have the value of "a", we can find the common difference. Since the fifth term is 3a, the common difference is (3a - a) = 2a. So, the common difference is 2 * 12 = 24.
Step 6: Now we can find the sum of the arithmetic progression. The formula for the sum of an arithmetic series is: S = (n/2)(2a + (n-1)d), where "n" is the number of terms.
In this case, the number of terms is 5.
Plugging the values into the formula: S = (5/2)(2*12 + (5-1)*24)
Simplifying this equation, we get: S = (5/2)(24 + 4*24) = (5/2)(24 + 96) = (5/2)(120) = 300.
Therefore, the sum of the arithmetic progression is 300.
To find the sum of an arithmetic progression, we need to know the first term (a), the common difference (d), and the number of terms (n).
Let's start by finding the value of the fifth term (T5) in terms of the first term (T1). We are given that the fifth term is three times the first term, so we can write:
T5 = 3 * T1
Next, we are told that a quarter of the fifth term is 9. So, we can write this as an equation:
(1/4) * T5 = 9
Substituting the value of T5 from the first equation, we get:
(1/4) * (3 * T1) = 9
Simplifying this equation, we have:
(3/4) * T1 = 9
To isolate T1, we divide both sides of the equation by (3/4):
T1 = (9 * 4) / 3
T1 = 12
Now that we know the value of the first term (T1), we can find the common difference (d) by subtracting the first term from the second term. We know that the fifth term is three times the first term, so the second term would be T1 + d. Substituting the values, we have:
T5 = T1 + 4d (since the fifth term is four terms ahead of the first term)
We already found T1 to be 12, so we can write:
3 * T1 = 12 + 4d
Simplifying this equation, we get:
36 = 12 + 4d
Subtracting 12 from both sides of the equation, we have:
36 - 12 = 4d
24 = 4d
Dividing both sides by 4, we find:
6 = d
Now we know the first term (T1 = 12) and the common difference (d = 6).
Finally, we can find the sum of the arithmetic progression using the formula:
Sn = (n/2)(2a + (n-1)d)
Substituting the values, we get:
Sn = (n/2)(2 * 12 + (n-1) * 6)
The number of terms (n) is not given in the question, so we cannot calculate the sum without this information. Please provide the number of terms in the arithmetic progression to find the sum.