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(Properties of Arithmetic Progression)`
1. If `a_1, a_2 , a_3, ...` are in AP with common difference d, then
`a_1 ± k, a_2 ± k, a_3 ± k, ...` are also in AP with common difference d.
2. If `a_1, a_2 , ...` are in AP with common difference d, then
`a_1k , a_2k , a_3k ,...` and `a_1/k , a_2/k , a_3/k , ..........` are also in `AP (k ne 0)` with common
differences `kd` and `d/k` respectively.
3. If `a_1, a_2 , ..` and `b_1, b_2 , b_3 , ..` are two `AP's` with common differences `d_1`
and `d_2`, respectively. Then, `a_1 pm b_1 , a_2 pm b_2 , a_3 pm b_3,...........` are also in AP with
common difference `(d_1 ± d_2 ).`
4. If `a_1, a_2 , a_3 , ...` and `b_1 , b_2 , b_3 ,....`. are two `AP' s` with common differences `d_1`
and `d_2` respectively, then `a_1b_1,a_2b_2,...........` and `a_1/b_1 , a_2/b_2,............` are not in
AP.
5. If `a_1, a_2 , a_3 ... , a_n` are in AP, then
`a_r = (a_(r-k) + a_(r+k))/2 , AA k, 0 <= k <= n -r`
6. If three numbers in AP whose sum is given are to be taken as `alpha-beta, alpha, alpha+beta`
and if five numbers in AP whose sum is given, are to be taken as
`alpha- 2 beta , alpha - beta , alpha , alpha + beta, alpha+ 2 beta` etc.
`text(In general)` , If (2r + 1) numbers in AP whose sum is given, are to be
taken as `(r in N).`
`alpha - rbeta , alpha- ( r-1)beta ..........., alpha - beta , alpha , alpha + beta , ....... , alpha + (r-1) beta, alpha + r beta`
`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`.
`(vii)` If a be the first term and d be the common difference, then AP can be written as
`a,a+ d,a+2d, ... ,a+ (n-1)d, ... ,` where `n in N. `
`(viii)` If we add the common difference to any term of AP, we get the next following term and if we subtract it from any term, we get the preceding term.
`(ix)` The common difference of an AP may be positive, zero, negative or imaginary.
`(x)` Constant AP common difference of an AP is equal to zero.
`(text(xi))` Increasing AP common difference of an AP is greater than zero.
`(text(x ii ))` Decreasing AP common difference of an AP is less than zero.
`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.
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(Properties of Arithmetic Progression)`
1. If `a_1, a_2 , a_3, ...` are in AP with common difference d, then
`a_1 ± k, a_2 ± k, a_3 ± k, ...` are also in AP with common difference d.
2. If `a_1, a_2 , ...` are in AP with common difference d, then
`a_1k , a_2k , a_3k ,...` and `a_1/k , a_2/k , a_3/k , ..........` are also in `AP (k ne 0)` with common
differences `kd` and `d/k` respectively.
3. If `a_1, a_2 , ..` and `b_1, b_2 , b_3 , ..` are two `AP's` with common differences `d_1`
and `d_2`, respectively. Then, `a_1 pm b_1 , a_2 pm b_2 , a_3 pm b_3,...........` are also in AP with
common difference `(d_1 ± d_2 ).`
4. If `a_1, a_2 , a_3 , ...` and `b_1 , b_2 , b_3 ,....`. are two `AP' s` with common differences `d_1`
and `d_2` respectively, then `a_1b_1,a_2b_2,...........` and `a_1/b_1 , a_2/b_2,............` are not in
AP.
5. If `a_1, a_2 , a_3 ... , a_n` are in AP, then
`a_r = (a_(r-k) + a_(r+k))/2 , AA k, 0 <= k <= n -r`
6. If three numbers in AP whose sum is given are to be taken as `alpha-beta, alpha, alpha+beta`
and if five numbers in AP whose sum is given, are to be taken as
`alpha- 2 beta , alpha - beta , alpha , alpha + beta, alpha+ 2 beta` etc.
`text(In general)` , If (2r + 1) numbers in AP whose sum is given, are to be
taken as `(r in N).`
`alpha - rbeta , alpha- ( r-1)beta ..........., alpha - beta , alpha , alpha + beta , ....... , alpha + (r-1) beta, alpha + r beta`
`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`.
`(vii)` If a be the first term and d be the common difference, then AP can be written as
`a,a+ d,a+2d, ... ,a+ (n-1)d, ... ,` where `n in N. `
`(viii)` If we add the common difference to any term of AP, we get the next following term and if we subtract it from any term, we get the preceding term.
`(ix)` The common difference of an AP may be positive, zero, negative or imaginary.
`(x)` Constant AP common difference of an AP is equal to zero.
`(text(xi))` Increasing AP common difference of an AP is greater than zero.
`(text(x ii ))` Decreasing AP common difference of an AP is less than zero.
`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.