Find the sum of the first 25 terms of an arithmetic sequence whose 7th term is −247 and whose 18th term is −49.
7th term = -247 ---> a+6d = -247
18th term = -49 ---> a+17d =-49
subtract them:
11d = 198
d = 18
in a+6d = -247
a + 108 = -247
a = -355
sum(25) = (25/2)(-710 + 24(18)) = -3475
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To find the sum of the first 25 terms of an arithmetic sequence, we need to know the first term, the common difference, and the last term. However, we are only given the values of the 7th and 18th terms.
To find the common difference (d), we can subtract the 7th term from the 18th term:
d = (-49) - (-247)
d = -49 + 247
d = 198
With the common difference, we can find the first term (a) by subtracting 6 times the common difference from the 7th term:
a = (-247) - 6d
a = (-247) - 6(-198)
a = (-247) + 1188
a = 941
Now that we have the first term (a = 941), the common difference (d = 198), and the number of terms (n = 25), we can use the formula for the sum of an arithmetic sequence:
Sum = (n/2)(2a + (n-1)d)
Substituting the given values, we have:
Sum = (25/2)(2(941) + (25-1)(198))
Sum = (25/2)(1882 + 24(198))
Sum = (25/2)(1882 + 4752)
Sum = (25/2)(6634)
Sum = 25*3317
Sum = 82925
Therefore, the sum of the first 25 terms of the arithmetic sequence is 82925.