Sequences & Series
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Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Arithmetic Sequences
Identifying arithmetic sequences
A sequence is arithmetic if the difference between consecutive terms is constant, ie un+1−un=d, the common difference, for all n. For example
is arithmetic but
is not since 9−5=4=5−2=3.
General term
The nth term in an arithmetic sequence is given by
where u1 is the first term and d is the common difference.
Geometric Sequences
Identifying Geometric Sequences
A sequence is geometric if the ratio between consecutive terms is always constant, i.e.
We call r the common ratio.
For instance, the sequence
is geometric with r=41, but
is not since 69=23=2=36.
General Term of a Geometric Sequence
The nth term of a geometric sequence is given by
where u1 is the first term and r is the common ratio.
Arithmetic Series
A series is the sum of a sequence
The sum of terms in a sequence is called a series.
Calculating arithmetic series
The sum of the first n terms in an arithmetic sequence is given by
or equivalently
Σ summation notation
Understanding summation notation
As a shortcut for writing out long sums, we can use the symbol ∑ with the following "settings":
Here n is called the index, but other letters can also be used in place of n.
Properties of Σ
Sums with ∑ have the following properties:
Sum of a sum:
Sum of a multiple:
Sum of constant:
Splitting the sum:
Sums with scalar multiples
Sum of a constant
Splitting a sum in Σ form
For any series of the form k=1∑nak and any integer m between 1 and n, we can split the series at the index m:
Geometric Series
Finite Geometric Series
The sum of the first n terms in a geometric sequence is given by:
Convergence
A geometric series is said to converge if S∞ is finite - which means ∣r∣<1⇔−1<r<1.
Example
A geometric sequence has u1=8 and u4=2k+1. For what value(s) of k does the corresponding geometric series converge?
We have
Now if −1<r<1, then −1<r3<1:
Infinite Geometric Series
If a geometric sequence has a common ratio ∣r∣<1, then each term will be smaller than the previous term. As the terms get smaller and smaller, the sum of all the terms approaches a finite value:
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Compounding (Appreciation & Depreciation)
Depreciation
where FV is the future value, PV is the present value, n is the number of years, and r% is the annual depreciation rate of the item.
Compound Interest Formula
where FV is the future value, PV is the present value, n is the number of years, k is the number of compounding periods per year, and r% is the nominal annual rate of interest.
Inflation & Real Value
The real interest rate (needed when a question involves inflation) is given by r%=c%−i%, where c% represents the given interest rate (the nominal rate) and i% represents the inflation rate.
Note: You can calculate the real interest rate r% and enter it directly into the TVM solver (when required) as the nominal annual interest rate (I% on your calculator), since the TVM solver does not account for inflation effects in its standard calculations.
Positive & Negative Cash Flows (TVM)
Whenever you use the Finance App (TVM Solver) on your calculator, it's critical that you enter and interpret the signs correctly:
When you receive money from a bank or savings account, that value is positive, because you're gaining money.
When you send money to a bank, that value is negative, because you're losing money.
Using TVM Solver (Calculator) - Compound Interest
You should understand the meaning of each variable and know how to use your calculator's Finance/TVM Solver:
To solve for an unknown, move your calculator's cursor to the unfilled slot and press alpha → enter.
Be very careful if P/Y is different from C/Y. The letter N will always be the number of payment periods, or in other words the number of years times P/Y.