In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference.
Solution
Let the first term be \(a\) and the common difference be \(d\).
The fifth term is given by:
\(a + 4d = 197\)
The sum of the first ten terms is given by:
\(\frac{10}{2} [2a + 9d] = 2040\)
Simplifying the sum equation:
\(5(2a + 9d) = 2040\)
\(2a + 9d = 408\)
We now have two equations:
1. \(a + 4d = 197\)
2. \(2a + 9d = 408\)
Multiply the first equation by 2:
\(2a + 8d = 394\)
Subtract this from the second equation:
\((2a + 9d) - (2a + 8d) = 408 - 394\)
\(d = 14\)
Thus, the common difference is 14.
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