4.3.1 Adding Consecutive Terms of Arbitrary Fibonacci Sequence, Method 1

This is a common question where the test writer gives the beginning and end terms of an Arbitrary Fibonacci Sequence and asks for the sum of a subset of the terms.

The Derivation

Using the telescoping properties of the Fibonacci recursion relation $A_n = A_{n-1} + A_{n-2}$:

$$A_1 + A_2 + A_3 + \dots + A_n = A_{n+2} - A_2$$

The Rule

The sum of the first $n$ terms of an Arbitrary Fibonacci Sequence is: $A_{n+2} - A_2$

Example: Sum of 4, 7, 11, …, 47, 76

We need to find $A_9 - A_2$ (since we have 7 terms, n=7, so we need $A_9$):

Term	Value
A₆	47
A₇	76
A₈	47 + 76 = 123
A₉	76 + 123 = 199

Sum = 199 − 7 = 192

Tip: This method is best when the test writer explicitly writes most of the sequence, so you only need to compute 2-3 additional terms.

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Problem Set 4.3.1

Practice these problems. Type your answer and press Enter to check: