PROGRESSION & SERIES
· ·. . · ARITHMETIC PROGRESSION . · .
· Show that the sequence defined by tn=4n+5 is an A.P.
Show that the sequence defined by tn=2n 2+1 is not an A.P.
G) Show that any sequence is an A.P ..if its nth term is a linear expression in n and the
common difference is the coefficient of n.
th
The nth term of a sequen~e is 3n-2. Is the sequence an A.P.? Find its 10 term. 28
In the A.P. 2,5,8, .... upto 50 terms and 3,5,7,9 ...... upto 60 terms, find how many
terms are identical. 20
If a1, a 2, a 3,...... an are in A.P. where ai>0 for all i. Show that
1 1 1 n-1
J;; +Ta;+- +Jan=
+ ...... + Jan-1 {a; +..ja;. •
If a1, a2, a3, .... an be non-zero terms of an A.P.; prove that
1 1 , i n-1
- + - - + ...... + - - = - - ·
a1a2 a2a3 an-1an a1an
If a1, a2, a3, .. .... an be non-zero terms of an A.P ., prove that
_J_ + _1_ + _1_ + ..... ; + _1___2_ (_!_; J_ + ..... + J_)
8 18 n a2 8 n-1 a3an-2 a1an a1 + an a1 a2 an
a
Find the sum of first 24 terms of the A.P. 1, a2, a3, ..... an _if a1+as+a10+a1s+a2o+a24=225.900
:1
,;
Solve 1+6+11+16+.: ... +x=148
If the first term of an A.P. is 2 and the sum of first five terms is equal to one fourth of
the sum of next fi.ve terms, find the sum of first 30 terms.
36
-2550
;$ . If in an A.P. the sum of first rn terms is equal to n and the sum of first n terms is m,
,JY
then prove that the sum of (m+n) terrfis is '.'"'(m+n).
If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show
that the sum of its (m+n) terms is zero. .,
::·14. If S1,S 2 ,S3, ..... Sm are the sums of n terms of m A.P.'s whose first terms are
1,2,3, ..... ,m and common difference are 1,3,5, .... ,(2m-1) respectively. Show that
mn .
S1+S2+ ....... +Sm= (mn+ 1) .
2
',
y. The ratio of the sum of n terms .of two AP's is (7n+1):(4n+27) . Find the ratio of their
14m-6 ·
mth terms Sm+ 23 1
2 2
The ratio• qf th~ sums of m and n terms of an A.P. is m :n . Show that the ratio of the
mth and nth terms is (2m-1):(2n-1) '
17. If the sum of first m terms of an A.P. is equal to the sum of next n terms and also
I equal to the sum of next p terms; prove that (m+n2(.!. -
r ··
!)
m p
= (m +pf.!. -
If there are (2n+1) terms in A.P., then prove that the ratio of the sum of odd terms and
\m n
!)
j
the sum of even terms is (n+1):n.
y The interior angles of a polygon are in A.P. The . smallest angle is 120° and the
common difference is 5°. Find the number of sides of the polygon. 9
, G For what value of th e parameter a are there real values of x such that
5 ,~ + 5 1 ~ , a ,2sx + 25~ are three consecutive terms of an A.P.
2 ae[12,oc)
~1. If the sum of four integers in A.P. is 24 a_nd their product is 945, find the numbers. 3,5,7,9
fl. If a,, 8 2, a3 ,-----------------------.:------------------, a2" are in A.P; show that
a,2 -a; +a; -a; + ......... +a22 -1 -:a22 =-n-.-(a2 -a2) .
n .' n 2n - I I 2n
23. Find four distinct integers in A.P if one of these is the sum of the squares of the
remaining three. -1, 1, o, 2
24. Balls are arranged in rows to form an equilateral triangle. The first row consists of one
ball, the second row of 2 balls a~d so on . If 669 more balls are added, then all the balls
can be arranged in the shape of a sc1uare and each of its sides contains 8 balls less than
each side of the triangle did. Determine the initial nu.mber of balls. 1540
25. If the pth term of an A.P is x and qth term is y, show that the sum of (p+q) terms is
p+q
. 2
(x + y + p-q
x- YJ
The number of terms of an A.P is even . The sum of odd terms is 24 and of even terms is
. · . 21 · .
. 30. If the_last term exceeds the first term by then find number of terms: 8
2
27. 150 workers were engaged to finish a piece of work in a certain number of days. Four
workers dropped the second day, four more workers dropped the third day and so ·on. It
takes 8 more days to finish the work now. Find the number of days in which the work ·was
completed. 25
28. Along a ·road lie an odd num_ ber of stones at intervals of 10 metres. These stones have to :
be assembled around the f!1iddle stone. A person can carry only one ston.e at a time. A
man carried the job with one · of the end stones by carrying tttem in succession. In
carrying all the stones he covered a distance of 3km. Find the number of stones. 25
. · ·. · · . ARITHMETIC MEAN ·.
Insert three arithmetic means between 3 and 19. 7,11,15 ·
n+1 bn+1
For what value of n, a + is the AM between a and b. 0
an+ bn .
3
Tlie sum of two nu·mbers is ~ • An even number of arithmetic means are being
inserted between them and their sum exceeds their number by 1. Find the number of
. . . .12
means inserted . .
If the AM between pth and qth terms of an AP be equal to the AM between_ rth and sth ·.
terms of the AP, then -~ how that p+q=J+s . th · · th
Between 1 and 31; n A.M's are ~nserted such that the ratio of th~ 7 and (n:- 1\
1
means is 5:9, find n. .· ·· · . · . 4
Find the relation between x and y iryorder th.at rth mean between x a~d 2y may be
same as rth mean between 2x and y if n means are inserted in each case. . ·
· .· · · . . ry=(n+1-r) x •J
{
· ·. . · ARITHMETIC PROGRESSION . · .
· Show that the sequence defined by tn=4n+5 is an A.P.
Show that the sequence defined by tn=2n 2+1 is not an A.P.
G) Show that any sequence is an A.P ..if its nth term is a linear expression in n and the
common difference is the coefficient of n.
th
The nth term of a sequen~e is 3n-2. Is the sequence an A.P.? Find its 10 term. 28
In the A.P. 2,5,8, .... upto 50 terms and 3,5,7,9 ...... upto 60 terms, find how many
terms are identical. 20
If a1, a 2, a 3,...... an are in A.P. where ai>0 for all i. Show that
1 1 1 n-1
J;; +Ta;+- +Jan=
+ ...... + Jan-1 {a; +..ja;. •
If a1, a2, a3, .... an be non-zero terms of an A.P.; prove that
1 1 , i n-1
- + - - + ...... + - - = - - ·
a1a2 a2a3 an-1an a1an
If a1, a2, a3, .. .... an be non-zero terms of an A.P ., prove that
_J_ + _1_ + _1_ + ..... ; + _1___2_ (_!_; J_ + ..... + J_)
8 18 n a2 8 n-1 a3an-2 a1an a1 + an a1 a2 an
a
Find the sum of first 24 terms of the A.P. 1, a2, a3, ..... an _if a1+as+a10+a1s+a2o+a24=225.900
:1
,;
Solve 1+6+11+16+.: ... +x=148
If the first term of an A.P. is 2 and the sum of first five terms is equal to one fourth of
the sum of next fi.ve terms, find the sum of first 30 terms.
36
-2550
;$ . If in an A.P. the sum of first rn terms is equal to n and the sum of first n terms is m,
,JY
then prove that the sum of (m+n) terrfis is '.'"'(m+n).
If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show
that the sum of its (m+n) terms is zero. .,
::·14. If S1,S 2 ,S3, ..... Sm are the sums of n terms of m A.P.'s whose first terms are
1,2,3, ..... ,m and common difference are 1,3,5, .... ,(2m-1) respectively. Show that
mn .
S1+S2+ ....... +Sm= (mn+ 1) .
2
',
y. The ratio of the sum of n terms .of two AP's is (7n+1):(4n+27) . Find the ratio of their
14m-6 ·
mth terms Sm+ 23 1
2 2
The ratio• qf th~ sums of m and n terms of an A.P. is m :n . Show that the ratio of the
mth and nth terms is (2m-1):(2n-1) '
17. If the sum of first m terms of an A.P. is equal to the sum of next n terms and also
I equal to the sum of next p terms; prove that (m+n2(.!. -
r ··
!)
m p
= (m +pf.!. -
If there are (2n+1) terms in A.P., then prove that the ratio of the sum of odd terms and
\m n
!)
j
the sum of even terms is (n+1):n.
y The interior angles of a polygon are in A.P. The . smallest angle is 120° and the
common difference is 5°. Find the number of sides of the polygon. 9
, G For what value of th e parameter a are there real values of x such that
5 ,~ + 5 1 ~ , a ,2sx + 25~ are three consecutive terms of an A.P.
2 ae[12,oc)
~1. If the sum of four integers in A.P. is 24 a_nd their product is 945, find the numbers. 3,5,7,9
fl. If a,, 8 2, a3 ,-----------------------.:------------------, a2" are in A.P; show that
a,2 -a; +a; -a; + ......... +a22 -1 -:a22 =-n-.-(a2 -a2) .
n .' n 2n - I I 2n
23. Find four distinct integers in A.P if one of these is the sum of the squares of the
remaining three. -1, 1, o, 2
24. Balls are arranged in rows to form an equilateral triangle. The first row consists of one
ball, the second row of 2 balls a~d so on . If 669 more balls are added, then all the balls
can be arranged in the shape of a sc1uare and each of its sides contains 8 balls less than
each side of the triangle did. Determine the initial nu.mber of balls. 1540
25. If the pth term of an A.P is x and qth term is y, show that the sum of (p+q) terms is
p+q
. 2
(x + y + p-q
x- YJ
The number of terms of an A.P is even . The sum of odd terms is 24 and of even terms is
. · . 21 · .
. 30. If the_last term exceeds the first term by then find number of terms: 8
2
27. 150 workers were engaged to finish a piece of work in a certain number of days. Four
workers dropped the second day, four more workers dropped the third day and so ·on. It
takes 8 more days to finish the work now. Find the number of days in which the work ·was
completed. 25
28. Along a ·road lie an odd num_ ber of stones at intervals of 10 metres. These stones have to :
be assembled around the f!1iddle stone. A person can carry only one ston.e at a time. A
man carried the job with one · of the end stones by carrying tttem in succession. In
carrying all the stones he covered a distance of 3km. Find the number of stones. 25
. · ·. · · . ARITHMETIC MEAN ·.
Insert three arithmetic means between 3 and 19. 7,11,15 ·
n+1 bn+1
For what value of n, a + is the AM between a and b. 0
an+ bn .
3
Tlie sum of two nu·mbers is ~ • An even number of arithmetic means are being
inserted between them and their sum exceeds their number by 1. Find the number of
. . . .12
means inserted . .
If the AM between pth and qth terms of an AP be equal to the AM between_ rth and sth ·.
terms of the AP, then -~ how that p+q=J+s . th · · th
Between 1 and 31; n A.M's are ~nserted such that the ratio of th~ 7 and (n:- 1\
1
means is 5:9, find n. .· ·· · . · . 4
Find the relation between x and y iryorder th.at rth mean between x a~d 2y may be
same as rth mean between 2x and y if n means are inserted in each case. . ·
· .· · · . . ry=(n+1-r) x •J
{
