Sequences and Series HSC ADVANCED MATHEMATICS
How do you find the formula for the nth Term of an arithmetic sequence or series?
For an ARITHMETIC sequence or series, substitute into the general formula Tₙ = a + (n-1) d, the values in the specific sequence for a and the difference ,d. This can be used to find any term in that sequence or series. The term formula arithmetic is on the formula sheet provided by NESA
What is the nth partial sum?
The sum formula used to find the sum of the first n terms
How do I evaluate a series?
This means to find the sum of all the terms given. You can just add them up or you can use the sum formula which is on the formula sheet provided by NESA
What is the test for an arithmetic sequence or series?
T₃ - T₂ = T₂ - T₁ and this is the common difference, d and T₁ is the first term, a
What is the test for a geometric sequence or series?
T₃/T₂ = T₂/T₁ and this is the common ratio, r and T₁ is the first term, a
How can you find a term formula if you are given the Sum formula?
Tₙ = Sₙ - Sₙ₋₁ If you substitute (n-1) into the Sₙ formula and then subtract it from the Sₙ formula, you will be left with the formula for Tₙ
If you know two terms in a series or sequence, how do you find the two variables needed to generate the whole sequences or series?
Use the general term formula and substitute the value of the term into Tₙ and use the value of n for each term to find two equations and then solve simultaneously for the variables needed
How do you find out if a number belongs to a sequence or series?
Use the term formula, Tₙ. Substitute the number that you are checking in as Tₙ and find if n is a positive counting number. If n is not a positive integer then the value that you used for Tₙ in NOT in the sequence or series
How do you find the first positive term in a sequences or series?
Make Tₙ > 0 and solve for n. REMEMBER, n is a counting number so it is always a positive integer. Check by substituting in the n value that you have decided on
How do you find the number of terms in a series presented in sigma notation form?
Subtract the first value for n from the last value of n and +1 How many terms in this series? 10 - 5 + 1 = 6 terms in this series
How do you handle sequences and series that have log terms?
Simplify the logs using the log rules then use the tests to determine if it is an arithmetic or geometric relationship between the terms
What is the limiting Sum?
When you add terms in a GEOMETRIC SERIES where r is a fraction between -1 and +1, |r|<1, then the sum converges to a single value. This is found by using the formula S∞ = a/(1-r) which is on the NESA formula sheet
Why do we use Σ notation?
Σ is used to simplify a series. ALWAYS generate the series to understand the pattern. The term numbers are written below and above the sigma (Σ) and the nth term formula is written to the right of Σ
