4.3.3 Adding Odd of Even Terms of Arbitrary Fibonacci Sequence

Again, the derivations are pretty lengthy, so lets just look at the results.

For the sum of the odd terms (e.g., $A_1, A_3$, etc…) of an Arbitrary Fibonacci Sequence:

$$\sum_{i=1}^{n} A_{2i-1} = A_1 + A_3 + A_5 + \dots + A_{2n-1} = A_{2n} - (A_2 - A_1)$$

What this means is that the sum is equal to the next term in the complete sequence (which will be an even term) with the difference between the first and second terms subtracted from it. Using our previous example sequence of 4, 7, 11, 18, 29, 47, …, then the sum of the first 3 odd terms (4, 11, 29) is:

$$X = A_6 - (A_2 - A_1) = 47 - (7 - 4) = 44$$

For the sum of the even terms (e.g., $A_2, A_4$, etc…) of an Arbitrary Fibonacci Sequence:

$$\sum_{i=1}^{n} A_{2i} = A_2 + A_4 + A_6 + \dots + A_{2n} = A_{2n+1} - A_1$$

What this means is that the sum is equal to the next term in the complete sequence (which will be an odd term) with the first term subtracted from it. Using our previous example sequence of 4, 7, 11, 18, 29, 47, …, then the sum of the first 3 even terms (7, 18, 47) is:

$$X = A_7 - A_1 = 76 - 4 = 72$$

Now these are trivial examples where the sum is simple to compute. In order to use the formulas, you’ll need to either have a long list of terms given in the problem statement or they’ll ask about the Standard Fibonacci Sequence which you’d then have the next term in the sequence memorized to help with the calculations. For example:

The sum of the first 7 odd terms of the Standard Fibonacci Sequence: $X = F_1 + F_3 + \dots + F_{13} = F_{14} - (F_2 - F_1) = 377 - (1 - 1) = 377$

The sum of the first 5 even terms of the Standard Fibonacci Sequence: $X = F_2 + F_4 + \dots + F_{10} = F_{11} - F_1 = 89 - 1 = 88$

Now I haven’t explicitly seen any problems that use these sequences, but it wouldn’t hurt to be familiar with these procedures if you suddenly see these types of problems make an appearance on either the Number Sense or Mathematics exams.