Question

Total list of formulas in the chapter ARITHMETIC PROGRESSIONS(class10).

Answer 1

★★★ ARITHMETIC PROGRESSION ★★★

There are basically three types of Progression .
(1) Arithmetic progression
(2) Geometric progression
(3) Harmonic progression

But in 10th syllabus, you have only ARITHMETIC progression .

★ARITHEMATICS progression :-
A sequence , or progression is said to be arithmetic progression when the difference of two consequitive terms remains constant .

This constant is known as common difference . denoted by " d " .

if " a " is the first term and " d " is the common difference of an AP .

➡ nth term ( Tn) = a + ( n -1)d
➡Sum of n terms (Sn) = n/2{ 2a + ( n -1)d }
or ,
if 1st term and nth term given then,
Sum of nth term ( Sn) =
n/2{ 1st term + nth term }

SHORTCUT TIPS
================

(1) The sum of an AP consisting of odd number of terms = n( middle term)

(2) if * a " is the first term and d is common difference of an AP having m terms , then nth terms from end is ( m - n +1)th term from the beginning .
thus, nth term from the end = a + ( m -n)d
example :-
2, 4 , 6, ............20 are in AP series
find 7th term from the ends ?
here we see
total number of terms ( m ) = 10
n = 7
so, 7th term from the end = 2 + ( 10-7)2
=2 + 6 = 8

★SOME IMPORTANT FACTS
ABOUT an AP★
================================
➡ if fixed number is added to or subtracted from each term of a given AP , then resulting sequence is also an AP. and it has same common difference .
e.g
A1 , A2 , A3 , A4 , ......An are in AP.
then, after adding or subtracting K from each term of given series .

A1 ±K , A2 ±K, A3 ± K , A4± K , ........An±K are also in AP. and same common difference .

➡ if each term of an AP, multiplied or divided by fixed non- zero constant , then resulting sequence also in AP with common difference constant times of original AP.
e.g
A1, A2 , A3 , A4 .......An are in AP
common difference = ( A2 - A1) = d
now, A non -zero constant K multiplied or divided to each term of given AP.
A1× K, A2× K, A3 × K, .........An× K are in AP.
common difference = ( A2- A1)K = dK

A1/K, A2/K , A3/K , ........An/K are in AP.
common difference = (A2 - A1)/K = d/K

➡ if P1, P2 , P3 ......are in AP
and Q1 , Q2 , Q3 .... are also in AP.
then,
(1) P1 ± Q1 , P2 ±Q2 , P3 ± Q3 .....are in AP
(2) P1Q1, P2Q2 , P3Q3 .... are NOT in AP
(3) P1/Q1, P2/Q2, P3/Q3 ...are NOT in AP.

➡ if Sn given ,
Q ask find nth term of AP
then ,
nth term ( Tn) = Sn - S(n-1)
= sum of n terms - sum of (n -1)terms

➡ if nth term of a sequence is a linear expression in n then the sequence is an AP. e.g Tn = An + b

example :- Tn = 4n + 5 , Tn = 7n -5 etc are in AP.

➡ In case, if we have to make a choice of few terms which are in AP, then suggested choice is as follows ( specially when sum of term is given )
for odd
========
for three terms =>(a - d) , a , (a +d )
for five terms =>(a -2d), (a -d), a , (a +d), (a +2d)

for even
===========
for 2 terms => (a - d), (a + d)
for four terms => (a -3d), (a -d), (a +d ), (a +3d)

★★ ARITHMETIC MEAN ★★

➡ single arithmetic mean :- A number is said to be the single arithmetic mean between two given members a , and b
e.g A.M = ( a + b)/2

➡if A1 , A2 , A3, A4 , A5 .....An be the arithmetic mean between a, and b .
then,
Kth Arithmetic mean = a + Kd
where d = ( b -a)/( n +1)

[ note :- I gave AM concept .
by the way AM , rarely use in 10th class . you should give less effort for AM.
this is 11th class contents ]


[ I think , I gave approximately all concept including 10th Arithmetic progression. may be some formula , I forgot to give here. so, you should read books and if you go to class or coaching , must read sir assignment ]

Answer 2

Hi ,

If a is the first term and ' d ' is the

common difference of an A P . then

the general form of A. P is

a , a+ d , a + 2d , a + 3d , ....

i) nth term = an = l = a + ( n - 1 ) d

ii ) If S n is the sum of 'n ' terms of an

A.P whose first term is ' a ' , common

difference ' d ' then

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

b ) Sn = n/2 [ a + l ]

Where a is first term and. ' l ' is the

last term.

iii ) If a , x , b are in A.P then x is

called arithmetic mean between ' a '

and ' b '

x - a = b - x

2x = a + b

1 ) x = ( a + b ) / 2

2 ) The arithmetic mean between a and b is (a+ b )/2

3 ) If a , x1 , x2 , ......., xn , b are in A.P

then x1 , x2 , x3 , ......, xn are called ' n'

Arithmetic means between a and b.

4 ) If n A.M 's are inserted in between

a and b then common difference ( d )

Of the A.P = ( b - a )/ ( n+ 1) ;

5 ) nth A.M = ( a + nb)/ ( n+ 1)

6 ) The sum of ' n ' arithmetic means

= n/2 [a + b ]

I hope this will usful to you.

****