Question

The first fifteen Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 and 610. Get the sum of the first two terms and compare it with the fourth term. Get the sum of the first three terms and compare it with the fifth term. Get the sum of the first four terms and compare it with the sixth term. Without performing addition, what is the sum of the first thirteen terms of the Fibonacci numbers?

Answer 1

Solution :-

→ The first fifteen Fibonacci numbers are = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 and 610.

now,

→ sum of first 2 terms = 1 + 1 = 2

→ fourth term = 3 = Next term of sum of first 2 terms .

checking next,

→ sum of the first three terms = 1 + 1 + 2 = 4

→ fifth term = 5 = Next term of sum of first 3 terms .

again,

→ sum of the first four terms = 1 + 1 + 2 + 3 = 7

→ sixth term = 8 = Next term of sum of first 3 terms .

then, we can conclude that, In Fibonacci numbers ,

• sum of first 2 terms = 2 + 2 = 4th term .
• sum of first 3 terms = 3 + 2 = 5th term .
• sum of first 4 terms = 4 + 2 = 6th term .
• ____________________________ .
• ____________________________ .
• sum of first 13 terms = 13 + 2 = 15th term .

therefore,

→ The sum of the first thirteen terms of the Fibonacci numbers = 610 (Ans.)

Learn more :-

let a and b positive integers such that 90 less than a+b less than 99 and 0.9 less than a/b less than 0.91. Find ab/46

p...

https://brainly.in/question/40043888

Answer 2

Given :  The first fifteen Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 and 610.

To Find : Without performing addition,   the sum of the first thirteen terms of the Fibonacci numbers

Solution:

Sum of 1st two terms  = 1 + 1  = 2   =  3 - 1  = Fourth term - 1

Sum of 1st three terms  = 1 + 1  + 2 = 4   =  5 - 1  = Fifth term - 1

Sum of 1st four terms  = 1 + 1  + 2 + 3= 7  =  8 - 1  = Sixth term - 1

Hence

Sum of 1st n terms  = (n + 2)th term  -  1

Sum of 1st  13  terms  = (13 + 2)th term  -  1

=>Sum of 1st  13  terms  =15th term  -  1

=> Sum of 1st  13  terms  = 610  -  1

=>  Sum of 1st  13  terms  = 609

Without performing addition,   the sum of the first thirteen terms of the Fibonacci numbers  is 609

Verification :

1 +1+ 2+ 3+ 5+8+ 13+ 21+ 34+ 55+89+144+ 233  = 609

Additional Info :

Fibonacci

a₁ = 1

a₂ = 1

aₙ =  aₙ₋₁   + aₙ₋₂

Learn More

Fibonacci अनुक्रम निम्नलिखित रूप में परिभाषित है

https://brainly.in/question/9228080

प्रथम पाँच पद लिखिये, जिनका n वाँ पद दिया गया है ।

brainly.in/question/9240363