The correct answer is option (B):
4929
Let's analyze the given arithmetic series: 97 + 101 + 105 + ... + 221.
First, we need to determine the key characteristics of this series. We can see that:
* The first term (a) is 97.
* The common difference (d) is 101 - 97 = 4.
* The last term (l) is 221.
Next, we need to find the number of terms (n) in this series. We can use the formula for the nth term of an arithmetic sequence:
l = a + (n - 1) * d
Substituting the values we have:
221 = 97 + (n - 1) * 4
Solving for n:
221 - 97 = (n - 1) * 4
124 = (n - 1) * 4
124 / 4 = n - 1
31 = n - 1
n = 32
So, there are 32 terms in the series.
Now, we can find the sum (S) of the arithmetic series using the formula:
S = n/2 * (a + l)
Substituting the values we have:
S = 32 / 2 * (97 + 221)
S = 16 * 318
S = 5088
Therefore, the sum of the series 97 + 101 + 105 + ... + 221 is 5088.
Looking back, we made an arithmetic error. Let's recalculate the sum:
S = n/2 * (a + l)
S = 32/2 * (97 + 221)
S = 16 * 318
S = 5088
It seems like there's an error in the provided options, or a calculation mistake in arriving at the expected correct answer of 4929. Let's review our calculations again:
n=32. a=97. l=221
S = (n/2) * (a + l)
S = (32/2) * (97+221)
S = 16 * 318
S = 5088
Given the options, the closest answer to the *correctly calculated* sum is 5088, but this is not one of the provided options. Let's check the correctness of the answer of 4929
If the answer is 4929, we are given:
S = 4929
n = 32
a = 97
l = ?
Using the sum formula:
S = (n/2)*(a+l)
4929 = (32/2)*(97+l)
4929 = 16*(97+l)
4929/16 = 97+l
308.0625 = 97 + l
l = 308.0625 - 97
l = 211.0625
This value of l (last term) 211.0625 is not an element of the sequence which means the answer 4929 is incorrect.
Given the common difference of 4, the final term would have to be of the form of 97 + 4*m where m is an integer. Thus, it looks as though there may be an error in the given question or answers. We can still identify that 5088 is the calculated sum, whereas the "correct" answer is 4929.