Arthmetic Progression (A.P.) :
It is sequence in which the difference between any term and its just preceding term remains constant throughout. This constant is called the `text("common difference")` of the `A.P.` and is denoted by `'d'` generally.
`text(Standard appearance of an A.P.)` is
`a, (a+d),(a+2d)............(a+ (n-1) d)`
where `'a'` denotes the first term of the `AP`
`text(General term)`/ `n^(th)` `text(term/Last term of)` `A.P:`
It is given by `T_n= a + (n - 1)d`
where `a =` first term, `d =` common difference and `n =` position of the term which we require.
`text (Note:)` If `d > 0 =>` increasing A.P.
If `d < 0 =>` decreasing A.P.
If `d = 0 =>` all the terms remain same
`text(Highlights of an A.P. :)`
`(i)` If each term of an `A.P.` is increased, decreased, multiplied or divided by the same non zero number, then the resulting sequence is also an `AP.`
`(ii)` Three numbers in `AP` can be taken as `a - d , a, a + d` ; four numbers in `AP` can be taken as `a - 3d, a - d, a + d, a + 3d` ; five numbers in `AP` are `a - 2d, a - d , a, a + d,a + 2d` & six terms in `AP` are `a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d` etc.
`(iii)` The common difference can be zero, positive or negative.
`(iv)` The sum of the two terms of an `AP` equidistant from the beginning & end is constant and equal to the sum of first & last terms.
`(v)` For any series, `T_n = S_n- S_(n-1)` In a series if `S_n` is a quadratic function of `n` or `T_n` is a linear function of `n`, then the series is an `A.P.`
`(vi)` If `a, b, c` are in `A. P. => 2b = a +c`.
`text(Arithmetic Mean )(A.M.) :`
When three quantities are in `A.P.` then the middle one is called the Arithmetic Mean of the other two.
`e.g. ``a, b, c` are in `A.P.` then `'b'` is the `text(arithmetic mean)` between `'a'` and `'c'` and `a+ c = 2b.`
It is to be noted that between two given quantities it is always possible to insert any number of terms such that the whole series thus formed shall be in `A.P.` and the terms thus inserted are called the arithmetic means.
`text(Geometric Progression G.P.)`
In a sequence if each term (except the first non zero term) bears the same constant ratio with its immediately preceding term then the series is called a `G.P.` and the constant ratio is called the `text(common ratio.)` Standard appearance of a `GP.` is
`a,ar,ar^2,ar^3,.........,ar^(n-1)`
`text(General term/)``n^t(h)` `text(term/Last term of )``G.P.`
It is given by `T_n = a . r^(n-1)`
where `a=` first term, `r=` common ratio and `n =` position of the term which we required.
`text(Highlights of)` `G.P.` :
`(i)` If each term of a `GP` be multiplied or divided by the same non-zero quantity, the resulting sequence is also a `GP.`
`(ii)` If each term of a `G.P.` raised the same power then resulting sequence is also a `GP.`
`(iii)` Any `3` consecutive terms of a `GP` can be taken as `a//r, a, ar` ;
`quadquadquadquad` Any `4 ` consecutive terms of a `GP` can be taken as `a//r^3, a//r, ar, ar^3` & so on.
`(iv)` ln a finite `G.P.` the product of the terms equidistant from the beginning and the end are equal.
`a_1a_n=a_2a_(n-1)=a_3a_(n-2)=................`
`(v)` If `a, b, c` are in `GP => b^2 = ac`.
`text(Geometric Mean) (G.M.) :`
If `a, b, c` are three positive numbers in `G.P.` then `b` is called the `text(geometrical mean)` between `a` and `c` and `b^2= ac.` If `a` and `b` are two positive real numbers and `G` is the `G.M.` between them, then
`quadquadquadquadquadquadquadG^2 = ab`
Arthmetic Progression (A.P.) :
It is sequence in which the difference between any term and its just preceding term remains constant throughout. This constant is called the `text("common difference")` of the `A.P.` and is denoted by `'d'` generally.
`text(Standard appearance of an A.P.)` is
`a, (a+d),(a+2d)............(a+ (n-1) d)`
where `'a'` denotes the first term of the `AP`
`text(General term)`/ `n^(th)` `text(term/Last term of)` `A.P:`
It is given by `T_n= a + (n - 1)d`
where `a =` first term, `d =` common difference and `n =` position of the term which we require.
`text (Note:)` If `d > 0 =>` increasing A.P.
If `d < 0 =>` decreasing A.P.
If `d = 0 =>` all the terms remain same
`text(Highlights of an A.P. :)`
`(i)` If each term of an `A.P.` is increased, decreased, multiplied or divided by the same non zero number, then the resulting sequence is also an `AP.`
`(ii)` Three numbers in `AP` can be taken as `a - d , a, a + d` ; four numbers in `AP` can be taken as `a - 3d, a - d, a + d, a + 3d` ; five numbers in `AP` are `a - 2d, a - d , a, a + d,a + 2d` & six terms in `AP` are `a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d` etc.
`(iii)` The common difference can be zero, positive or negative.
`(iv)` The sum of the two terms of an `AP` equidistant from the beginning & end is constant and equal to the sum of first & last terms.
`(v)` For any series, `T_n = S_n- S_(n-1)` In a series if `S_n` is a quadratic function of `n` or `T_n` is a linear function of `n`, then the series is an `A.P.`
`(vi)` If `a, b, c` are in `A. P. => 2b = a +c`.
`text(Arithmetic Mean )(A.M.) :`
When three quantities are in `A.P.` then the middle one is called the Arithmetic Mean of the other two.
`e.g. ``a, b, c` are in `A.P.` then `'b'` is the `text(arithmetic mean)` between `'a'` and `'c'` and `a+ c = 2b.`
It is to be noted that between two given quantities it is always possible to insert any number of terms such that the whole series thus formed shall be in `A.P.` and the terms thus inserted are called the arithmetic means.
`text(Geometric Progression G.P.)`
In a sequence if each term (except the first non zero term) bears the same constant ratio with its immediately preceding term then the series is called a `G.P.` and the constant ratio is called the `text(common ratio.)` Standard appearance of a `GP.` is
`a,ar,ar^2,ar^3,.........,ar^(n-1)`
`text(General term/)``n^t(h)` `text(term/Last term of )``G.P.`
It is given by `T_n = a . r^(n-1)`
where `a=` first term, `r=` common ratio and `n =` position of the term which we required.
`text(Highlights of)` `G.P.` :
`(i)` If each term of a `GP` be multiplied or divided by the same non-zero quantity, the resulting sequence is also a `GP.`
`(ii)` If each term of a `G.P.` raised the same power then resulting sequence is also a `GP.`
`(iii)` Any `3` consecutive terms of a `GP` can be taken as `a//r, a, ar` ;
`quadquadquadquad` Any `4 ` consecutive terms of a `GP` can be taken as `a//r^3, a//r, ar, ar^3` & so on.
`(iv)` ln a finite `G.P.` the product of the terms equidistant from the beginning and the end are equal.
`a_1a_n=a_2a_(n-1)=a_3a_(n-2)=................`
`(v)` If `a, b, c` are in `GP => b^2 = ac`.
`text(Geometric Mean) (G.M.) :`
If `a, b, c` are three positive numbers in `G.P.` then `b` is called the `text(geometrical mean)` between `a` and `c` and `b^2= ac.` If `a` and `b` are two positive real numbers and `G` is the `G.M.` between them, then
`quadquadquadquadquadquadquadG^2 = ab`