the least integer n for which the sum 1+3+5+---+(2n+1) exceeds 9999 is
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Answer / Amit Kumar
Let's find the value of n such that the sum S = 1 + 3 + 5 + ... + (2n + 1) > 9999. We can use the formula for the sum of an arithmetic series, which is Sn = n * (a + l) / 2, where a is the first term and l is the last term. Since S > 9999, we have n * ((1 + (2n + 1)) / 2) > 9999. Solving for n, we get n * (2n + 3) / 2 > 9999, or 4n^2 + 6n - 19998 < 0. Using the quadratic formula, we find two possible integer values of n: n = (-b ± sqrt(b² - 4ac)) / 2a = (6 ± sqrt((6)² - 4 * 4 * (-19998))) / (2 * 4). The positive solution is approximately n = 497, so the least integer value for n that satisfies the condition is n=497.
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