Evaluate: 5 + 8 + 11 + 14 + … … +395 + 398.
the series is
132
∑(3k+2) = 3∑k + 2∑1
k=1
= 3*(132)(133)/2 + 2*132
= 26598
To evaluate the given expression, we need to find the sum of the arithmetic sequence.
First, let's find the common difference, which can be determined by subtracting any two consecutive terms.
Common difference (d) = 8 - 5 = 3
Now, we need to find the number of terms (n). The last term in the sequence is 398, and the first term is 5. We can use the formula for finding the nth term of an arithmetic sequence to find n:
Last term (a_n) = First term (a_1) + (n - 1) * Common difference (d)
398 = 5 + (n - 1) * 3
398 - 5 = 3n - 3
393 = 3n
n = 393/3
n = 131
Therefore, the number of terms in the sequence is 131.
To find the sum (S) of an arithmetic sequence, we can use the formula:
Sum (S) = (n/2) * (First term + Last term)
S = (131/2) * (5 + 398)
S = (131/2) * 403
S = 131 * 201.5
S ≈ 26386.5
Hence, the sum of the given arithmetic sequence is approximately 26386.5.
To evaluate the given series, it is an arithmetic sequence where the common difference is 3. To find the sum of an arithmetic series, we can use the formula:
Sn = (n/2)(a + l)
Where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term (a) is 5 and the last term (l) is 398. We need to find the number of terms (n).
To find the number of terms (n) in an arithmetic sequence, we can use the formula:
l = a + (n-1)d
Where l is the last term, a is the first term, n is the number of terms, and d is the common difference.
In this case, l = 398, a = 5, and d = 3. Plugging these values into the formula, we can solve for n:
398 = 5 + (n-1)(3)
393 = 3n - 3
396 = 3n
n = 132
Now that we know the number of terms (n), we can find the sum (Sn) using the formula mentioned earlier:
Sn = (n/2)(a + l)
Sn = (132/2)(5 + 398)
Sn = 66(403)
Sn = 26,598
Therefore, the sum of the given series is 26,598.