![]() The formula for the nth term of a geometric progression whose first term is a and common ratio is r is a n = ar n−1.Each successive term is obtained in a geometric progression by multiplying the common ratio to its preceding term.In general, the arithmetic sequence can be represented as a, a+d, a+2d, a+3d.In an arithmetic sequence and series, a is represented as the first term, d is a common difference, a n as the nth term, and n as the number of terms.The following points are helpful to clearly understand the concepts of sequence and series. Infinite series: S n = a/(1−r) for |r| 1 The various formulas used in geometric sequence are given below: Geometric sequenceįinite series: S n = a(1−r n)/(1−r) for r≠1, and S n = an for r = 1 Successive term – Preceding term or a n - a n-1 The various formulas used in arithmetic sequence are given below: Arithmetic sequence Formulas related to various sequences and series are explained below: Arithmetic Sequence and Series Formula These formulas are different for each kind of sequence and series. There are various formulas related to various sequences and series by using them we can find a set of unknown values like the first term, nth term, common parameters, etc. A series formed by using harmonic sequence is known as the harmonic series for example 1 + 1/4 + 1/7 + 1/10. ![]() Harmonic Sequence and SeriesĪ harmonic sequence is a sequence where the sequence is formed by taking the reciprocal of each term of an arithmetic sequence. ![]() The geometric progression can be of two types: Finite geometric progression and infinite geometric series. A series formed by using geometric sequence is known as the geometric series for example 1 + 4 + 16 + 64. Geometric Sequence and SeriesĪ geometric sequence is a sequence where the successive terms have a common ratio. A series formed by using an arithmetic sequence is known as the arithmetic series for example 1 + 4 + 7 + 10. The types of sequence and series are:Īn arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference. Maybe these having two levels of numbers to calculate the current number would imply that it would be some kind of quadratic function just as if I only had 1 level, it would be linear which is easier to calculate by hand.There are various types of sequences and series, in this section, we will discuss some special and most commonly used sequences and series. This gives us any number we want in the series. I do not know any good way to find out what the quadratic might be without doing a quadratic regression in the calculator, in the TI series, this is known as STAT, so plugging the original numbers in, I ended with the equation:į(x) = 17.5x^2 - 27.5x + 15. Then the second difference (60 - 25 = 35, 95-60 = 35, 130-95=35, 165-130 = 35) gives a second common difference, so we know that it is quadratic. = a ( 4 ) + 2 =a(4)+2 = a ( 4 ) + 2 equals, a, left parenthesis, 4, right parenthesis, plus, 2 = 9 =\goldD9 = 9 equals, start color #e07d10, 9, end color #e07d10Ī ( 5 ) a(5) a ( 5 ) a, left parenthesis, 5, right parenthesis = 7 + 2 =\blueD 7+2 = 7 + 2 equals, start color #11accd, 7, end color #11accd, plus, 2 = a ( 3 ) + 2 =a(3)+2 = a ( 3 ) + 2 equals, a, left parenthesis, 3, right parenthesis, plus, 2 ![]() ![]() = 7 =\blueD 7 = 7 equals, start color #11accd, 7, end color #11accdĪ ( 4 ) a(4) a ( 4 ) a, left parenthesis, 4, right parenthesis = 5 + 2 =\purpleC5+2 = 5 + 2 equals, start color #aa87ff, 5, end color #aa87ff, plus, 2 = a ( 2 ) + 2 =a(2)+2 = a ( 2 ) + 2 equals, a, left parenthesis, 2, right parenthesis, plus, 2 = 5 =\purpleC5 = 5 equals, start color #aa87ff, 5, end color #aa87ffĪ ( 3 ) a(3) a ( 3 ) a, left parenthesis, 3, right parenthesis = a ( 1 ) + 2 =a(1)+2 = a ( 1 ) + 2 equals, a, left parenthesis, 1, right parenthesis, plus, 2 = 3 =\greenE 3 = 3 equals, start color #0d923f, 3, end color #0d923fĪ ( 2 ) a(2) a ( 2 ) a, left parenthesis, 2, right parenthesis = a ( n − 1 ) + 2 =a(n\!-\!\!1)+2 = a ( n − 1 ) + 2 equals, a, left parenthesis, n, minus, 1, right parenthesis, plus, 2Ī ( 1 ) a(1) a ( 1 ) a, left parenthesis, 1, right parenthesis A ( n ) a(n) a ( n ) a, left parenthesis, n, right parenthesis ![]()
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