Question

in how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers? (a) 1 (b) 3 (c) 5 (d) 6 (e) 7

Answer 1

To find the number of ways to write 345 as the sum of an increasing sequence of two or more consecutive positive integers, set up an equation and evaluate different values of n.

Step-by-step explanation:

To find the number of ways to write 345 as the sum of an increasing sequence of two or more consecutive positive integers, we can determine the upper limit of the sequence. Let's assume the sequence starts with the number x. Since the sum of the first n positive consecutive integers can be expressed as n(n+1)/2, we set up the equation:

x + (x+1) + (x+2) + ... + (x+n-1) = 345

Expanding the equation and rearranging terms, we get:

n(2x + n - 1) = 2*345 = 690

From here, we can test different values of n to find the possible values of x that satisfy the equation. By evaluating n for different values, we find that there are 7 ways to write 345 as the sum of an increasing sequence of two or more consecutive positive integers.

Answer 2

To find the number of ways in which 345 can be written as the sum of an increasing sequence of two or more consecutive positive integers, we can consider the largest number in the sequence. By solving an equation, we can determine the possible values for N and the corresponding lengths of the sequences.

Step-by-step explanation:

To find the number of ways in which 345 can be written as the sum of an increasing sequence of two or more consecutive positive integers, we can consider the largest number in the sequence. Let's assume it is N. Since the sequence is increasing, the smallest number will be N - (length of the sequence - 1). So, we need to find the value of N that satisfies the equation: N(N+1)/2 - (N - (length of the sequence - 1))(N - (length of the sequence - 1) + 1)/2 = 345. By solving this equation, we can determine the possible values for N and the corresponding lengths of the sequences. In this case, there are 5 possible values for N, so the answer is (c) 5.