Simran has a jar of 100 marbles. On day 1 she adds 1 marble. On day 2 she takes out 2 marbles. On day 3 she adds 3 marbles. On day 4 she takes out 4 marbles. If she continues this pattern for 100 days, how many marbles would be left in the jar? \( \begin{array}{lllll}\text { (A) } 50 & \text { (B) } 60 & \text { (C) } 70 & \text { (D) } 80 & \text { (E) } 90\end{array} \)

To determine how many marbles will be left in the jar after 100 days, let's analyze the pattern. On odd days, Simran adds marbles equal to the day's number, while on even days, she takes out marbles equal to the day's number. For the first 100 days, we can calculate: - On odd days (1, 3, 5,..., 99): There are 50 odd days. The sum of the first 50 odd numbers can be calculated using the formula for the sum of an arithmetic series: \( \text{Sum} = n^2 \), where \( n \) is the number of terms. Thus, \( 50^2 = 2500 \). - On even days (2, 4, 6,..., 100): There are also 50 even days. The sum of the first 50 even numbers (2, 4, 6,..., 100) can be calculated using the formula: \( \text{Sum} = n(n + 1) \). Here, \( n = 50 \) gives us \( 50 \times 51 = 2550 \). Now we can find the total number of marbles left in the jar: - Initial marbles: 100 - Marbles added over 100 days: +2500 - Marbles taken out over 100 days: -2550 Calculating the final count: \[ 100 + 2500 - 2550 = 50 \] So, after 100 days, Simran has 50 marbles left in the jar. The answer is (A) 50.