Find the the sum of first 35 terms of the series whose pth term is (p/7) +2
clearly this is an arithmetic sequence. If p=1, the 1st term is this 1/7 + 2 = 15/7
and the common difference is 1/7
So, S35 = 35/2 (2 * 15/7 + 34 * 1/7) = ____
Sure! Let's calculate the sum of the first 35 terms of the series.
First, let's find the first term of the series, which is given by the formula for the p-th term: (p/7) + 2.
When p = 1, the first term is (1/7) + 2 = 2 + 1/7.
Now, let's find the second term, when p = 2: (2/7) + 2.
Third term, when p = 3: (3/7) + 2.
And so on, until we find the 35th term.
Now, I would love to write out all the terms, but sadly, I can't scroll down on this screen. So, instead, let's use some humor to simplify our calculation.
If we observe closely, we can see that each term is (p/7) + 2, which is really (p/7) + (2 * 1).
So, we have p/7 + p/7 + p/7 + ... 35 times, and also 2 * 1 + 2 * 1 + 2 * 1 + ... 35 times.
Now, let's simplify it even more by noticing that the sum of "p/7" repeated 35 times can be written as (35 * p)/7.
Similarly, the sum of "2 * 1" repeated 35 times can be written as 2 * (35 * 1).
So, now we have the sum: (35 * p)/7 + 2 * (35 * 1).
Let's plug in the values: p = 35 and 1 = 1.
The sum of the first 35 terms of this series is: (35 * 35)/7 + 2 * (35 * 1).
Calculating that, we get: 5 * 35 + 2 * 35 = 175 + 70 = 245.
Therefore, the sum of the first 35 terms of this series is 245. Hope that brings a smile to your face!
To find the sum of the first 35 terms of the series, we need to calculate the sum of each term from p = 1 to p = 35.
Given that the pth term is defined as (p/7) + 2, we can replace p with the values from 1 to 35 and calculate the sum.
Let's calculate the sum of the first 35 terms step by step:
Step 1: Calculate the first term (p = 1):
First term = (1/7) + 2 = 1/7 + 14/7 = 15/7
Step 2: Calculate the second term (p = 2):
Second term = (2/7) + 2 = 2/7 + 14/7 = 16/7
Step 3: Calculate the third term (p = 3):
Third term = (3/7) + 2 = 3/7 + 14/7 = 17/7
Step 4: Calculate the fourth term (p = 4):
Fourth term = (4/7) + 2 = 4/7 + 14/7 = 18/7
Continue this pattern until the 35th term.
Step 5: Calculate the 35th term (p = 35):
35th term = (35/7) + 2 = 5 + 2 = 7
Now, we can calculate the sum of the first 35 terms.
Sum of the first 35 terms = (15/7) + (16/7) + (17/7) + ... + (7)
To find the sum, we can use the formula for the sum of an arithmetic series: Sn = (n/2) * (a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, we have:
n = 35 (since there are 35 terms)
a = 15/7 (first term)
l = 7 (35th term)
Sum = (35/2) * [(15/7) + 7]
= (35/2) * [(15 + 7*7)/7]
= (35/2) * [(15 + 49)/7]
= (35/2) * (64/7)
= (35 * 64) / (2 * 7)
= 2240 / 14
= 160
Therefore, the sum of the first 35 terms of the given series is 160.
To find the sum of the first 35 terms of the series, we will first calculate the value of each term and then add them up.
The pth term of the series is given by the formula:
T(p) = (p/7) + 2
To find the sum of the first 35 terms, we need to substitute the values of p from 1 to 35 into the formula and add them up.
Let's calculate the value of each term and then find the sum.
T(1) = (1/7) + 2 = 1/7 + 14/7 = 15/7
T(2) = (2/7) + 2 = 2/7 + 14/7 = 16/7
T(3) = (3/7) + 2 = 3/7 + 14/7 = 17/7
and so on...
To calculate T(35), we substitute p = 35 into the formula:
T(35) = (35/7) + 2 = 35/7 + 14/7 = 49/7
Now, let's find the sum of these 35 terms:
Sum = T(1) + T(2) + T(3) + ... + T(35)
= (15/7) + (16/7) + (17/7) + ... + (49/7)
= (15 + 16 + 17 + ... + 49) / 7
To find the sum of the numbers from 15 to 49, we can use the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
Here, the first term is 15, the last term is 49, and the number of terms is 35.
Sum = (35/2)(15 + 49) / 7
= (35/70)(64) / 7
= (1/2)(64) / 7
= 32 / 7
Therefore, the sum of the first 35 terms of the series is 32/7.