Question

Seventeen consecutive positive integers have a sum of 306. What is the sum of the seventeen consecutive integers that immediately follow the seventeen positive integers that sum to 306?

a) 323
b) 340
c) 357
d) 374

The mean of the list is 11. What is the median?
n,n+3,n+4,n+5,n+6,n+8,n+10,n+12,n+15
a) n+6
b) n+8
c) n+10
d) n+12

The arithmetic sequences 1, 5, 9, 13,... and 1, 8, 15, 22,... have infinitely many terms in common. Calculate the sum of the first three common terms.
a) 15
b) 18
c) 21
d) 24

Find the sum of the first 100 terms of the arithmetic sequence with the first term 2 and the common difference 5.
a) 25,000
b) 50,000
c) 75,000
d) 100,000

Answer 1

The sum of the seventeen consecutive integers immediately following a set that sums to 306 is 595. The median of the defined list of numbers is n+10. The sum of the first three common terms of the given arithmetic sequences is 87, and the sum of the first 100 terms of the sequence starting with 2 and a common difference of 5 is 24,950.

Step-by-step explanation:

For the first question, the sum of the next seventeen consecutive integers will be 17 times larger than the largest integer in the original set. Therefore, if the sum of the original set of seventeen consecutive integers is 306, we can find the mean by dividing by 17, which is 18. The largest integer is then 18 + 8 (since we have 17 numbers, 8 less than the mean and 8 more than the mean). This largest integer is 26, so the next set starts at 27. The sum is 306 + 17*17 (since we're adding 17 to each of the original numbers), which is 306 + 289 = 595. Therefore, the answer to the first question is 595.

The median of a list of numbers is the middle number when they are listed in order. From the list provided, n+6 is the 5th number and n+15 is the 13th number. Since the 9th number would be the median, and it falls within these two, the median is n+10.

The common terms in the given arithmetic sequences occur at multiples of the least common multiple (LCM) of the differences, which are 4 and 7. The LCM of 4 and 7 is 28. Thus, the common terms will be 1 (the first term they share), then 1 + 28 and so on. So the first three common terms are 1, 29, and 57. Their sum is 87.

The sum of the first 100 terms of any arithmetic sequence can be found with the formula: sum = n/2 * (first term + last term), where n is the number of terms. For this sequence, the first term is 2, and the last term can be found by the formula last term = first term + (n-1)*d, where d is the common difference. So the last term is 2 + 99*5 = 497. The sum of the first 100 terms is then 100/2 * (2 + 497) = 50*499 = 24,950.