Let `a` be the first term, `l` be the last term and `d` be the common difference of a certain sequence in AP. Then, `nth` term of AP is `T_n=a+ (n -1) d =l`

where, `d=T_n-T_(n-1) , n ge 2 , n in N`

`nth` term from last, `T_n' = l-(n-1) d`

`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.

`S_n=a+(a+d)+(a+2d)+.......................+(a+(n-1) d)`

`S_n=(a+(n-1) d)+(a+(n-2)d)+(a+d)+...........................+a`

`2S_n=n[2a+(n-1)d]`

`S_n=n/2[2a+(n-1)d]`

or

`S_n=n/2[a+a+(n-1)d]`

`=> S_n=n/2(a+l)` where last term, `l=a+(n-1)d`

`text(Important Results :)`

1. If `S_n, t_n` and are sum of n terms, `n^(th)` term and common difference of an AP respectively then

`d = t_n - t_(n-1) `

`t_n = s_n - s_(n -1) `

`d = s_n- 2 s_(n-1) + s_(n- 2)`

2.. A sequence is an AP if and only if the sum of its `n` terms is of the form `An^2 + Bn,` where `A` and `B` are constants independent of `n.` In this case, the `nth` term and common difference of the `AP` are `A (2n - 1) +B` and `2A` respectively.

3. If `S_n = an^2 + bn + c,` where `S_n` denotes the sum of n terms of a series, then whole series is not an AP. It is AP from the second term onwards.

4. If ratio of the sums of `m` and `n` terms of an AP is given by

`S_m/S_n = (Am^2 + Bm)/(An^2 + Bn)`

If `a_1, a_2 , a_3 , . ..` are in AP, then

`=> a_1 ± k, a_2 ± k, a_3 ± k, ...` are also in AP.

`=> a_1k, a_2k, a_3 k, ...` and `a_1/k , a_2/k , a_3/k` are also in AP.



If `a_1 , a_2, a_3,......` and `b_1, b_2, b_3,.......` are two Ap's , then

`=> a_1 pm b_1, a_2 pm b_2, a_3 pm b_3,.....` are also in AP.

`=> a_1b_1, a_2 b_2, a_3 b_3,.....` and `a_1/b_1 , a_2/b_2, a_3/b_3....` are not in AP

If `a_1, a_2 , a_3 , ...` are in AP, then

`=> a_1 + a_n = a_2 + a_(n-1) = a_3 + a_(n-2)=......`

`=> a_r =(a_(r-k) + a_(r+k))/2 , AA k; 0 le k le n-r`



Three numbers in an AP can be taken as `a-d, a, a+ d`.

Four numbers in an AP can be taken as `a-3d,a-d,a+d,a+3d`

If `nth` term of any sequence is linear expression inn such that `t_n =an + b`, then sequence is an AP with common difference `a`.


If sum of `n` terms of any sequence is quadratic expression in `n` i.e. `S_n = an ^2 + bn`, then sequence is an AP.

If `a, b, c` are three positive numbers in `A.P.` then `b` is called the `text(arithmetic mean)` between `a` and `c` and `b= (a+c)/2.` If `a` and `b` are two positive real numbers and `A` is the `A.M.` between them, then
`quadquadquadquadquadquadquadb= (a+c)/2`

If `a, A_1 , A_2 , A_3 ............ , A_n , b` are in AP, then we say that `A_1 , A_2 , A_3 , ......... , A_n` arc the n arithmetic means (AM) between two numbers tl and b. The common difference `(d)` of this AP is `(b-a)/(n+1)` and `m th` arithmetic mean is given by `A_m = a+ (m (b-a))/(n+1)`

By putting `m = 1 , 2 , 3 , .............., n` we can get the values of `A_1 , A_2 , ........... , A_n` The sum of `n` arithmetic means between two given numbers is `n` times the single AM between them i.e. `A_1 +A_2+A_3 + ........+ A_n = n` (single AM between a and b) If there is only one arithmetic mean 'A' between `a` and `b`, then `A = (a+b)/2`

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.

`S = a + ar + ar^2 + .......... + ar^(n - 1)`

`S.r=+ar+ar^2+...............................+ar^n`

subtract

`S(1-r)=a-ar^n=a(1-r^n)`

`S=(a(1-r^n))/(1-r)` ,where `r ne 1` , (if `r=1` then `S=na`)

When `|r | < 1` and number of terms is infinite, then

`lim_(n-> oo) r^n =0`

`:. S_(oo) = a/(1-r)` where, `r= t_n/(t_(n-1))`

1. If `a_1 , a_2 , a_3 ........` are in GP with common ratio r, then `a_1k , a_2k , a_3k,........` and `a_1/k ,a_2/k ,a_3/k , ..........` are also in GP `(k ne 0)` with common ratio `r.`

2. If `a_1, a_2 , a_3, ...` are in GP with common ratio r, then `a_1 pm k , a_2 pm k , a_3 pm k , ...........` are not in GP `(k ne 0).`

3. If `a_1, a_2 , a_3, ...` are in GP with common ratio `r,` then

`(i) 1/a_1 , 1/a_2 . 1/a_3 ............` are also GP common ratio `1/r`
`(ii) a_1^n , a_2^n , a_3^n ,............` are also in GP with common ratio `r^n` and `n in Q.`
`(iii) loga_1 , loga_2 , loga_3 , .............` are in AP `(a_1 > 0, AA i)`
In this case, the converse also holds good.

4. If `a_1, a_2 , a_3, ...` and `b_1, b_2 , b_3, ...` are two GP's with common ratios `r_1` and `r_2,` respectively. Then,

`(i) a_1b_1 , a_2b_2 , a_3b_3 , ..........` and `a_1/b_1 , a_2/b_2 , a_3/b_3,.............` are also in GP with common ratios `r_1r_2` and `r_1/r_2` respectively.
`(ii)` `a_1 pm b_1 , a_2 pm b_2 , a_3 pm b_3 ,.............` are not in GP.


5. If three numbers in GP whose product is given are to be taken as `a/r , a,ar` and if five numbers in GP whose product is given are to be taken as `a/r^2,a/r ,a , ar ,ar^2 ,..etc`

6. If four numbers in GP whose product is given are to be taken as `a/r^3 , a/r , ar , ar^3` and if six numbers in GP whose product is given are to be taken as `a/r^5 , a/r^3 , a/r , ar , ar^3, ar^5,` etc.

Let `a_1, a_2 , ...., a_n` be in AP and `b_1, b_2 , .... , b_n` be in GP.

Then, `a_1b_1, a_2b_2 , ... , a_nb_n` will be in AGP. If the first term of an AP is a, common difference is `d` and first term of a GP is b,
common ratio is `r`, then

`ab, (a+ d) br, (a+ 2d) br^2, ....` are in AGP.

Now, `S_n =(ab)/(1-r) + (dbr (1-r^(n-1)))/((1-r)^2) - ([a+(n-1)d]br^n)/(1-r), r ne 1`

If `-1 < r < 1` then `lim_(n->oo) r^n=0`

and `lim_(n->oo) nr^n=0`

`:.` Sum to infinity , `S_(oo) =(ab)/(1-r) + (dbr)/((1-r)^2)`

If `a,x_1 ,x_2 , ... ,x_n ,b` are in HP, then `x_1 ,x_2 , .. ,x_n` are in harmonic means between `a` and `b`.

These are the reciprocal of n arithmetic means between `1/a` and `1/b` .

Hence, `x_1 =((n+1)ab)/(a+nb), x^2 = ((n+1))/(2a +(n-1)b),........,` and so on.