The Fibonacci sequence is defined by 1=a1=a2 and an= an 1+an 2, n > 2. Find an+1an for n=1,2,3,4,5 (2024)

FAQs

What sequence is defined as a1 a2 1 and an an 1 an 2? ›

The Fibonacci sequence is defined by a1 = 1 = a2, an = an–1 + an–2 for n > 2.

The Fibonacci sequence is defined by 1=a1=a2 and an= an 1+an 2, n > 2.

1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 233 , 377 , ... F n = F n − 2 + F n − 1 where n ≥ 2 . Each term of the sequence , after the first two, is the sum of the two previous terms. This sequence of numbers was first created by Leonardo Fibonacci in .

The Fibonacci sequence begins with the following 14 integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... Each number, starting with the third, adheres to the prescribed formula. For example, the seventh number, 8, is preceded by 3 and 5, which add up to 8.

Hence, the above sequence doesn't form an A.P.

Write the first five terms in each of the following sequence: a1 = a2 =2, an = an–1 – 1, n > 2 - Noon Academy. Answer: Using n = 1, 2, 3, 4, 5, the first five terms can be calculated. ∴ First five terms of the sequence are 2, 2, 1, 0, -1.

Fibonacci Sequence Formula

The kick-off part is F₀=0 and F₁=1. The recursive relation part is F_n = F_n-1+F_n-2. It is noted that the sequence starts with 0 rather than 1. So, F₅ should be the 6^th term of the sequence.

The Fibonacci sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... In this series, the next number is found by adding the two numbers before it. Hence, the next term in the series is 8 + 13 = 21.

In the pattern of triangular numbers you will see, the next number in the sequence is added with an extra row. Let us explain in detail. So the sequence formed here is in the pattern: 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, and so on.

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by 12 gives the next term. In other words, an=a1rn−1 a n = a 1 r n - 1 . This is the form of a geometric sequence.

Is 1 1 2 3 5 8 a geometric sequence? ›

It is neither geometric nor arithmetic. Not all sequences are geometric or arithmetic. For example, the Fibonacci sequence 1,1,2,3,5,8,... is neither. A geometric sequence is one that has a common ratio between its elements.

The sequence's general term or nth term is calculated using the formula, an=a1+(n-1) d where a1 is the first term, d is a common difference, n is the term position, and an is the nth term of the sequence.

The difference between each successive term is the same. Therefore, the given list of numbers form an AP.

This is an arithmetic sequence since there is a common difference between each term.

∴ The above sequence is not an AP. Was this answer helpful? Check whether an=2n2+1 is an AP or not.