aboslutelydestroying

\(\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10}\) is basically:

1/3(1 - 1/4 + 1/4 -1/7 +1/7 - 1/10)

which is 1/3 \(\times\) 9/10 = 3/10

If \(S_1\) is 1/2

that means \(a_1\) is 1/2.

If \(a_1 \) is 1/2, then \(a_2\) is 0

So \(a_{15}\) is 1/2 - 14(1/2) = 13/2

So 1/2 + 13/2 = 7

7 times 15/2 = 105/2

1% x 200 = 2

so 2kg of berries did not change.

198kg were water.

10% of the water were gone.

10% x 200 = 20

That means now 178kg were now water

So 178+2 = 180

The 23rd term is equal to: \(a_1\times r^{22}\)

The 26th terms is equal to: \(a_1\times r^{25}\)

So then \(r^3= \)\(12\over{16}\)

r^3 = 3/4

r = \(\sqrt[3]{3\over4}\)

32nd term would be 23rd term times r^9

\((\sqrt[3]{3\over4})^9\)=3^3/4^3 = 27/64

16\(\times\)27/64 = 27/4

She starts at 2, so she at least needs 98 more to go.

The only number that is divisible by 5 and is the smallest number greater than 98 is 100

So Laverne sends Shirley screaming and running at 2 + 100 = 102

Thanks

The quadratic equation takes the form of x^2 - mx + 24 = 0. Given that x_1 and x_2 are its roots and are integers, it means that the sum of these roots (x_1 + x_2) equals m (based on Vieta's formulas), and the product of these roots equals 24 (x_1*x_2=24). In this case, we need to factor the constant term (24), in other words, list all possible pairs of integers that multiply to give 24. These include {1,24}, {-1,-24}, {2,12}, {-2,-12}, {3,8}, {-3,-8}, {4,6}, {-4,-6}, and their negative counterparts. As both roots are integers, and each pair of factors correspond to one unique value of m (the sum of those two integers), there are altogether 16 different values for m, considering both positive and negative pairs.

Answer is: 16