Download ARITHMETIC SERIES. FORMULAE FOR THE NTH TERM AND

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ARITHMETIC SERIES. FORMULAE FOR THE NTH TERM AND SUM OF N TERMS
GENERAL SERIES
Series: the sum of a sequence of n terms
Term: the position of a number in a sequence, e.g. the first term is the first number in the sequence
Any general or nth term of a series is Tn where n stands for the number of the term and must be a positive integer.
For t h e series 6 + 13 + 20 + … fi n d
(a) T 1
(b) T 2
(c) T 3
(d ) T n
Solu t ion
(a)
(b)
(c)
(d )
Th e fi rst t erm is 6, so wh en n = 1: T 1 = 6
Th e 2 n d t erm is 13 so T 2 = 13
Th e 3 rd t erm is 20 so T 3 = 20
Each t erm is 7 m ore t h an t h e p reviou s t erm .
Th e 1 st t erm is 6
Th e 2n d t erm is 13 = 6 + 7
Th e 3 rd term is 20 = 6 + 7 + 7
= 6 + 2 ´7
SIGMA NOTATION
Sigma notation: Greek letter standing for the partial sum
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ARITHMETIC PROGRESSIONS (AP)
Arithmetic sequence: a set of numbers that form a pattern where each successive term is a constant amount (positive or negative) more than the
previous one
Common difference: the constant amount in an arithmetic sequence that is added to each term to get the next term, d
The first term is called a
Terms of an arithmetic series
Partial sum
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GEOMETRIC SERIES. FORMULAE FOR THE NTH TERM AND SUM OF N TERMS
GEOMETRIC PROGRESSIONS (GP)
Geometric sequence: a set of numbers that form a pattern where each successive term is multiplied by a constant amount to get the next term
Common ratio: the constant amount in a geometric sequence that is multiplied to each term to get the next term
The first term is called a
Terms of a geometric series
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GEOMETRIC SERIES WITH A RATIO BETWEEN -1 AND 1. THE LIMIT OF
, AS
, FOR
, AND THE
CONCEPT OF LIMITING SUM FOR A GEOMETRIC SERIES.
Partial sum
Limiting sum (sum to infinity
In some geometric series, the sum becomes very large as n increases, e.g. 2, 4, 8, 16. These series have an infinite sum.
In some geometric series, the sum doesn’t increase greatly after a few terms. These series have a limiting sum.
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APPLICATIONS OF ARITHMETIC SERIES. APPLICATIONS OF GEOMETRIC SERIES: COMPOUND INTEREST ,
SIMPLIFIED HIRE PURCHASE AND REPAYMENT PROBLEMS. APPLICATIONS TO RECURRING DECIMALS.
General applications
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Compound interest: interest is added to the balance of a bank account so that the interest and balance both earn interest
Money that earns compound interest grows according to a GP
Be sure that n & r
correspond, e.g. if n is in
months, then r must be
per month
Annuities: fixed sums of money invested every year that accumulate interest over a
number of years
Superannuation: a sum of money that is invested every year (or more frequently) as
part of a salary to provide a large amount of money when a person retires from the paid
workforce
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Loan repayments
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