The sequence {ak}112,k=1 satisfies a1=1 and an=1337+n/an−1, for all positive integers n. Let
S=⌊a10a13+a11a14+a12a15+⋯+a109a112⌋.
Find the remainder when S is divided by 1000.
explain, please?
To find the remainder when S is divided by 1000, we first need to understand the pattern in the given sequence {ak}112,k=1.
The given sequence is defined as a1=1 and an=1337+n/an−1, for all positive integers n.
To find any term in the sequence, we need to know the previous term. Let's calculate the first few terms to identify a pattern:
a1 = 1
a2 = 1338/1 = 1338
a3 = 1338/1338 = 1
a4 = 1337/1 = 1337
a5 = 1337/1337 = 1
...
From the calculations, we can see that the sequence alternates between 1 and 1337. This pattern continues for the remaining terms.
To simplify the expression S=⌊a10a13+a11a14+a12a15+⋯+a109a112⌋, we can replace the terms with their respective values:
S = ⌊(1)*(1) + (1337)*(1) + (1)*(1) + (1337)*(1) + ... (1)*(1337)⌋
We can notice that each term in the expression alternates between 1 and 1337. Therefore, the sum of these terms will have the same pattern.
The number of terms in the expression is 100, and since each term alternates between 1 and 1337, the sum can be simplified as follows:
S = (1 + 1337 + 1 + 1337 + ... + 1337) * (1 + 1337)
To find the sum of the terms with a common ratio, we can use the formula for the sum of an arithmetic series:
S = (n/2) * (first term + last term)
= (100/2) * (1 + 1337)
= 50 * 1338
= 66900
Now, to find the remainder when S is divided by 1000, we take the remainder of 66900 divided by 1000:
Remainder = 66900 % 1000
= 900
Therefore, the remainder when S is divided by 1000 is 900.