Question

The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, ... where the first and second terms are 1 and each term after that is the sum of the previous two terms. What is the remainder when the 100th term of the sequence is divided by 8?

Answer 1

The remainder when the 100th term of the Fibonacci sequence is divided by 8 is 3.

Step-by-step explanation:

The Fibonacci sequence is a sequence where each term is the sum of the previous two terms. The sequence starts with 1, 1, and then continues with 2, 3, 5, and so on. To find the remainder when the 100th term of the sequence is divided by 8, we can calculate the terms of the sequence modulo 8 until we reach the 100th term.

1. First term: 1 mod 8 = 1
2. Second term: 1 mod 8 = 1
3. Third term: (1 + 1) mod 8 = 2
4. Fourth term: (1 + 2) mod 8 = 3
5. Fifth term: (2 + 3) mod 8 = 5
6. Sixth term: (3 + 5) mod 8 = 0
7. Seventh term: (5 + 0) mod 8 = 5
8. Eighth term: (0 + 5) mod 8 = 5
9. Ninth term: (5 + 5) mod 8 = 2
10. Tenth term: (5 + 2) mod 8 = 7

We can see from the pattern that the terms modulo 8 repeat every 6 terms. Therefore, to find the remainder when the 100th term is divided by 8, we can calculate the remainder when 100 is divided by 6. 100 mod 6 = 4. So, the remainder when the 100th term of the Fibonacci sequence is divided by 8 is the same as the remainder when the 4th term is divided by 8, which is 3.

Answer 2

44,278,106,022,432,739,384.375 = 1/8 of ((1+√5)/2)^100/√5

Step-by-step explanation:

We separate the x10^21 number into 2 so that becomes x10^8 number with decimal of .75 when it is a x10^21 split into 2 large parts and a small subtraction creates the third step. we have found 1/4 then we half.

354224848179261915075 / 4

= 000000000000065478768.75

354224 848179000000000/4

-88556212044.8

= 354 has become a 12 digit number temporary starting with 885 the decimal .8 is simply put in front

3542248481 79- so it has become a x10^20 number. 885562120448,65478768.75

We arrange the comma's

88,556,212,044,865,478,768.75

88,556,212,044,865,478,768.75

88,556,212,044,865,478,768.75

88,556,212,044,865,478,768.75 is division by 4

=354224848179261915075

88,556,212,044,865,478,768.75 /2

= 44278106022 (/2) 432739384.375(/2) in two part.

= 44278106022,432739384.375 replace /2 with comma

= 44,278,106,022,432,739,384.375 replace all other comma.