progression & series

7/27/2019 progression & series

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Brain Teasures Progression and Series By Abhijit kumar Jha

http://akj259.worpress.com

EXERCISEI

Q.1 If the 10th term of an HP is 21 & 21st term of the same HP is 10, then find the 210th term.

Q.2 Show that ln (4 12 36 108 .............. up to n terms) = 2nln 2 + 3n2

)1n(nl

Q.3 There are n AMs between 1 & 31 such that 7th mean : (n 1)th mean = 5 : 9, then find the value of n.

Q.4 Find the sum of the series , 7 + 77 + 777 + ..... to n terms.

Q.5 Express the recurring decimal 0.1576 as a rational number using concept of infinite geometric series.

Q.6 Find the sum of the n terms of the sequence1

1 1 1

2

1 2 2

3

1 3 32 4 2 4 2 4

........

Q.7 The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369. The first

and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic

progression. Find the seventh term of the geometric progression.

Q.8 If the pth, qth & rth terms of an AP are in GP . Show that the common ratio of the GP isq r

p q

.

Q.9 If one AMa

& two GMs p & q be inserted between any two given numbers then show that

p3+ q3 = 2apq .

Q.10 The sum of n terms of two arithmetic series are in the ratio of (7n+ 1) : (4n+ 27) . Find the ratio of their

nth term.

Q.11 If S be the sum , P the product & R the sum of the reciprocals of a GP , find the value of PR

S

n

2

.

Q.12 The first and last terms of an A.P. are a and b. There are altogether (2n + 1) terms. A new series is

formed by multiplying each of the first 2n terms by the next term. Show that the sum of the new series is

n6

ab)2n4()ba)(1n4( 2222 .

Q.13 In an AP of which a is the Ist term, if the sum of the Ist p terms is equal to zero , show that the sum of

the next q terms is a(p+q)q/(p1).

Q.14(a) The interior angles of a polygon are in AP. The smallest angle is 120 & the common difference is 5.

Find the number of sides of the polygon.

(b) The interior angles of a convex polygon form an arithmetic progression with a common difference of 4.

Determine the number of sides of the polygon if its largest interior angle is 172.

Q.15 An AP & an HP have the same first term, the same last term & the same number of terms ; prove that

the product of the rth term from the beginning in one series & the rth term from the end in the other is

independent of r.

Q.16 Find three numbers a, b, c between 2 & 18 such that ;(i) their sum is 25

(ii) the numbers 2, a, b are consecutive terms of an AP &

(iii) the numbers b, c

, 18 are consecutive terms of a GP .

Q.17 Given that ax = by = cz = du & a, b

, c

, d are in GP, show that x

, y

, z

, u are in HP .

Q.18 In a set of four numbers, the first three are in GP & the last three are in AP , with common difference 6.

If the first number is the same as the fourth , find the four numbers.
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Q.19 Find the sum of the first n terms of the sequence : 1 2 11

3 11

4 11

2 3

n n n.........

Q.20 Find the nth term and the sum to n terms of the sequence :

(i) 1+5+13+29+61+ ...... (ii) 6+13+22+33+ .......

Q.21 The AM of two numbers exceeds their GM by 15 & HM by 27 . Find the numbers.

Q.22 The harmonic mean of two numbers is 4. The airthmetic mean A & the geometric mean G satisfy the

relation 2A + G = 27. Find the two numbers.

Q.23 Sum the following series to n terms and to infinity :

(i)1

1 4 7

1

4 7 10

1

7 10 13. . . . . . ......... (ii)

r

n

1

r (r+1)(r+2) (r+3)

(iii)r

n

1

1

4 12

r (iv) ...........

8.6.4

5.3.1

6.4

3.1

4

1 .

Q.24 Find the value of the sum

(a)r

n

1 s

n

1

rs

2r 3s where rs

is zero if r s & rs

is one if r = s.

(b)i

n

j

i

k

j

1 1 1

1.

Q.25 For or 0 < < /2, if :

x =n

0

cos2n , y =n

0

sin2n , z =n

0

cos2n sin2n then : Prove that

(i) xyz = xy + z (ii) xyz = x + y + z

EXERCISEII

Q.1 The series of natural numbers is divided into groups (1), (2, 3, 4), (5, 6, 7, 8, 9), ...... & so on. Show

that the sum of the numbers in the nth

group is (n

1)3

+ n3

.Q.2 The sum of the squares of three distinct real numbers , which are in GP is S . If their sum is S, show

that (1/3,1) (1,3) .

Q.3 If there be m APs beginning with unity whose common difference is 1, 2, 3 .... m . Show that the sum

of their nth terms is (m/2) (mn m+ n + 1).

Q.4 If Snrepresents the sum to n terms of a GP whose first term & common ratio are a & r respectively, then

prove that S1+ S

3+ S

5+ ..... + S

2n-1=

a n

r

a r r

r r

n

1

1

1 1

2

2

( )

( ) ( ).

Q.5 A geometrical & harmonic progression have the same pth, qth & rth terms a, b, c respectively. Show

that a(b c)loga+ b(c a)logb+ c(a b)logc = 0.Q.6 A computer solved several problems in succession. The time it took the computer to solve each successive

problem was the same number of times smaller than the time it took to solve the preceding problem.

How many problems were suggested to the computer if it spent 63.5 min to solve all the problems

except for the first, 127 min to solve all the problems except for the last one, and 31.5 min to solve all the

problems except for the first two?

Q.7 If the sum of m terms of an AP is equal to the sum of either the next n terms or the next p terms of the

same AP prove that (m+ n) [(1/m)(1/p)] = (m + p) [(1/m)(1/n)] (n p)


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Q.8 If the roots of 10x3 cx2 54x 27 = 0 are in harmonic progression, then find c & all the roots.

Q.9(a) Let a1, a

2, a

3...... a

nbe an AP . Prove that :

1 1 1 1

1 2 1 3 2 1a a a a a a a an n n n

.......... =2 1 1 1 1

1 1 2 3a a a a a an n

..........

(b) Show that in any arithmetic progression a1, a

2, a

3.......

a1 a

2 + a

3 a

4 + ...... + a

2K 1 a2K = [K/(2K 1)](a

1 a

2K) .

Q.10 Let a1

, a2

, ........... , an

, an+1

, ....... be an A.P.

Let S1

= a1

+ a2

+ a3

+ ............. + an

S2

= an+1

+ an+2

+ ...............+ a2n

S3

= a2n+1

+ a2n+2

+ .............+ a3n

................................................

................................................

Prove that the sequence S1

, S2

, S3

, ........ is an arithmetic progression whose common difference

is n2 times the common difference of the given progression.

Q.11 If a, b, c are in HP, b, c, d are in GP & c, d, e are in AP, Show that e = ab/(2ab) .

Q.12 If a, b, c, d, e be 5 numbers such that a, b, c are in AP ; b, c, d are in GP & c, d, e are in HP then:(i) Prove that a, c, e are in GP .

(ii) Prove that e = (2b a)/a .(iii) If a = 2 & e = 18 , find all possible values of b

, c

, d .

Q.13 The sequence a1, a

2, a

3, ....... a

98satisfies the relation a

n+1= a

n+ 1 for n = 1, 2, 3, .........97 and has

the sum equal to 4949. Evaluate

49

1kk2

a .

Q.14 If n is a root of the equation x(1ac) x(a+c) (1+ac) = 0 & if n HMs are inserted betweena & c, show that the difference between the first & the last mean is equal to ac(a c) .

Q.15 (a) The value of x + y + z is 15 if a, x

, y

, z , b are in AP while the value of ;

(1/x)+(1/y)+(1/z) is 5/3 if a, x

, y

, z

, b are in HP . Find a & b .

(b) The values of xyz is 15/2 or 18/5 according as the series a, x

, y

, z

, b is an AP or HP. Find

the values of a & b assuming them to be positive integer .

Q.16 An AP, a GP & a HP have a& b for their first two terms . Show that their (n+2)th terms will be

in GP if n

1n

abab

abn2n2

2n22n2

.

Q.17 Prove that the sum of the infinite series 23..........2

9.7

2

7.5

2

5.3

2

3.1

432 .

Q.18 If there are n quantities in GP with common ratio r & Sm

denotes the sum of the first m terms, show that

the sum of the products of these m terms taken two & two together is [r/(r+1)][Sm

][Sm 1] .


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Q.19 Find the condition that the roots of the equation x3px2 + qx r = 0 may be in A.P. and

hence solve the equation x312x2 + 39x 28 = 0.

Q.20 If ax2 + 2bx+ c = 0 & a1x2 + 2b

1x+ c

1= 0 have a common root & a/a

1, b/b

1, c/c

1are in AP,

show that a1

, b1

& c1

are in GP.

Q.21 If a, b

, c be in GP & log

ca, log

bc, log

ab be in AP , then show that the common difference of the

AP must be 3/2.

Q.22 If a1

= 1 & for n>1, an

= an-1

+1

1a n , then show that 12 < a

75< 15.

Q.23 Sum to n terms : (i)1

1

2

1 2

3

1 2 3

2

x

x

x x

x

x x x

( ) ( ) ( ) ( ) ( ).......

(ii)

a

a

a

a a

a

a a a

1

1

2

1 2

3

1 2 31 1 1 1 1 1

.......

Q.24 In a GP the ratio of the sum of the first eleven terms to the sum of the last eleven terms is 1/8 and the

ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is 2.

Find the number of terms in the GP.

Q.25 Given a three digit number whose digits are three successive terms of a G.P. If we subtract 792 from it,

we get a number written by the same digits in the reverse order . Now if we subtract four from the

hundred's digit of the initial number and leave the other digits unchanged, we get a number whose digitsare successive terms of an A.P. Find the number.

EXERCISEIII

Q.1 For any odd integer n 1, n3 (n 1)3 + ...... + (1)n 1 l3 = ______ . [ JEE 96, 1]

Q.2 x = 1+ 3a + 6a + 10a3 + ..... a < 1y = 1+ 4b + 10b + 20b3 + .....b < 1, find S = 1+ 3ab + 5(ab) + .... in terms of x & y.

[ REE 96, 6 ]

Q.3 The real numbers x1, x

2, x

3satisfying the equation x3 x+ x+ = 0 are in A.P . Find the

intervals in which and lie . [JEE96, 3]

Q.4 Let p & q be roots of the equation x2 2x + A = 0, and let r & s be the roots of the equationx2 18x + B = 0 . If p < q < r < s are in arithmatic progression, then A = ______, and B = ______.

[ JEE 97, 2 ]

Q.5 a, b, c are the first three terms of a geometric series. If the harmonic mean of a & b is 12 and that of b

& c is 36, find the first five terms of the series. [ REE '98, 6 ]

Q.6 Select the correct alternative(s). [ JEE '98, 2 + 2 + 8 ]

(a) Let Trbe the rth term of an AP, for r = 1, 2, 3, .... If for some positive integers m, n we have

Tm

=1

n

& Tn

=1

m

, then Tmn

equals :

(A)1

mn(B)

1 1

m n (C) 1 (D) 0

(b) If x = 1, y > 1, z > 1 are in GP, then1

1 n x,

1

1 n y,

1

1 n zare in :

(A) AP (B) HP (C) GP (D) none of the above

(c) Prove that a triangle ABC is equilateral if & only if tanA+ tanB+ tanC = 3 3 .


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Q.7(a) The harmonic mean of the roots of the equation 5 2 x2 4 5 x + 8 + 2 5 = 0 is(A) 2 (B) 4 (C) 6 (D) 8

(b) Let a1, a

2,...., a

10, be in A.P. & h

1, h

2, ....., h

10be in H.P. If a

1= h

1= 2 & a

10= h

10= 3 then a

4h

7is:

(A) 2 (B) 3 (C) 5 (D) 6

[ JEE '99, 2 + 2 out of 200 ]

Q.8 The sum of an infinite geometric series is 162 and the sum of its first n terms is 160. If the inverse of its

common ratio is an integer, find all possible values of the common ratio, n and the first terms of the series.

[ REE '99, 6 ]

Q.9(a) Consider an infinite geometric series with first term 'a' and common ratio r . If the sum is 4 and the

second term is 3/4, then :

(A) a =7

4, r =

3

7(B) a = 2 , r =

3

8

(C) a =3

2, r =

1

2(D) a = 3 , r =

1

4

(b) If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies

the relation :

(A) 0 M 1 (B) 1 M 2(C) 2 M 3 (D) 3 M 4

[ JEE 2000, Screening, 1 + 1 out of 35 ]

(c) The fourth power of the common difference of an arithmetic progression with integer entries added to

the product of any four consecutive terms of it . Prove that the resulting sum is the square of an integer.

[ JEE 2000, Mains, 4 out of 100 ]

Q.10 Given that , are roots of the equation, Ax2 4x + 1 = 0 and , the roots of the equation,Bx2 6x + 1 = 0, find values of A and B, such that , , & are in H.P.

[ REE 2000, 5 out of 100 ]

Q.11 The sum of roots of the equation ax2 + bx + c = 0 is equal to the sum of squares of their reciprocals. Find

whether bc2, ca2 and ab2 in A.P., G.P. or H.P.? [ REE 2001, 3 out of 100 ]

Q.12 Solve the following equations for x and y

log2x + log

4x + log

16x + .................... = y

5 9 13 4 1

1 3 5 2 1

............ ( )

.............. ( )

y

y= 4log

4x [ REE 2001, 5 out of 100 ]

Q.13(a) Let be the roots of x2x + p = 0 and be the roots of x24x + q = 0. Ifare in G.P.,then the integral values of p and q respectively, are(A) 2, 32 (B) 2, 3 (C) 6, 3 (D) 6, 32

(b) If the sum of the first 2n terms of the A.P. 2, 5, 8, ........... is equal to the sum of the first n terms of the A.P.

57, 59, 61, ........, then n equals

(A) 10 (B) 12 (C) 11 (D) 13

(c) Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are

(A) NOT in A.P./G.P./H.P. (B) in A.P.

(C) in G.P. (D) H.P.

[ JEE 2001, Screening, 1 + 1 + 1 out of 35 ]


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(d) Let a1, a

2.......... be positive real numbers in G.P. For each n, let A

n, G

n, H

n, be respectively, the

arithmetic mean, geometric mean and harmonic mean of a1, a

2, a

3, ...........a

n. Find an expression for the

G.M. of G1, G

2, .........G

nin terms of A

1, A

2.............A

n, H

1, H

2, .........H

n.

[ JEE 2001 (Mains); 5]

Q.14(a) Suppose a, b, c are in A.P. and a2, b2, c2 are in G.P. If a < b < c and a + b + c =3

2, then the value of

a is

(A)1

2 2

(B)1

2 3

(C)1

2

1

3

(D)1

2

1

2

[JEE 2002 (Screening), 3]

(b) Let a, b be positive real numbers. If a , A1

, A2

, b are in A.P. ; a , a1

, a2

, b are in G.P. and

a , H1

, H2

, b are in H.P. , show that

G G

H H

A A

H H

a b a b

ab

1 2

1 2

1 2

1 2

2 2

9

( ) ( )[ JEE 2002 , Mains , 5 out of 60 ]

Q.15 If a, b, c are in A.P., a2 , b2 , c2 are in H.P. , then prove that either a = b = c or a, b,2

c form a G.P..

[JEE-03, Mains-4 out of 60]

Q.16 The first term of an infinite geometric progression is x and its sum is 5. Then

(A) 0 x 10 (B) 0 < x < 10 (C) 10 < x < 0 (D) x > 10

[JEE 2004 (Screening)]

Q.17 If a, b, c are positive real numbers, then prove that [(1 + a) (1 + b) (1 + c)]7 > 77 a4 b4 c4.

[JEE 2004, 4 out of 60]

Q.18(a) In the quadratic equation ax2 + bx + c = 0, if = b24ac and + , 2 + 2, 3 + 3 are in G.P. where

, are the roots of ax2

+ bx + c = 0, then(A) 0 (B) b = 0 (C) c = 0 (D) = 0

[JEE 2005 (Screening)]

(b) If total number of runs scored innmatches is

4

1n(2n+1n 2) where n > 1, and the runs scored in

the kth match are given by k2n+1k, where 1 k n. Find n. [JEE 2005 (Mains), 2]

Q.19 If n

1n32

n 4

31.......

4

3

4

3

4

3A

and Bn = 1 AAn, then find the minimum natural

number n0 such that Bn > An. n > n0. [JEE 2006, 6]


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ANSWER KEY

EXERCISEI

Q 1. 1 Q 3. = 14

Q 4. S = (7/81){10n+1 9n 10} Q 5. 35/222Q 6. n(n+1)/2(n + n + 1) Q 7. 27

Q 10. (14n 6)/(8n + 23) Q 11. 1Q 14. (a) 9 ; (b) 12 Q 16. a = 5 , b = 8 , c = 12

Q 18. (8

,

4

, 2

, 8) Q 19. nQ 20. (i) 2n+1 3 ; 2n+2 4 3n (ii) n+ 4n + 1 ; (1/6)n (n+1)(2n + 13) + nQ 21. 120 , 30 Q 22. 6,3

Q 23. (i) sn

= (1/24) [1/{6(3n+ 1)(3n + 4)}] ; s = 1/24 (ii) (1/5)n (n+ 1)(n+ 2)(n+ 3)(n+ 4)

(iii) n/(2n+1) (iv) Sn

=

)2n2)(n2......(6.4.2

)1n2)(1n2.....(5.3.1

2

12 ; S = 1

Q 24. (a) (6/5)(6n 1) (b) [n(n+1)(n+2)]/6

EXERCISEII

Q 6. 8 problems , 127.5 minutes Q.8 C = 9 ; (3, 3/2 , 3/5)Q 12. (iii) b = 4, c = 6, d = 9 OR b = 2, c = 6 , d = 18 Q.13 2499Q 15. (a) a = 1, b = 9 OR b = 1, a = 9 ; (b) a = 1 ; b = 3 or vice versa

Q.19 2p3 9pq + 27r = 0; roots are 1, 4, 7

Q 23. (a) 1 x

x x x n

n

( ) ( ) ..... ( ) 1 2 (b) 1 1

1 1 11 2( ) ( ) ..... ( ) a a a n

Q 24. n = 38 Q 25. 931

EXERCISEIII

Q 1.1

4(2n 1)(n+1)

Q 2. S =1

12

ab

ab( )Where a = 1 x1/3 & b = 1 y1/4 Q3. (1/3) ; (1/27)

Q 4. 3, 77 Q 5. 8, 24, 72, 216, 648Q 6. (a) C (b) B Q 7. (a) B (b) D

Q 8. r = 1/9 ; n = 2 ; a = 144/180 OR r = 1/3 ; n = 4 ; a = 108 OR r = 1/81 ; n = 1 ; a = 160

Q 9. (a) D (b) A Q 10. A = 3 ; B = 8

Q 11. A.P. Q 12. x = 2 2 and y = 3

Q 13. (a) A, (b) C, (c) D , (d)

A A A H H H

n n

n

1 2 1 2

1

2, ,............ , ,............

Q14. (a) D Q.16 B Q.18 (a) C, (b) n = 7 Q.19 n0

= 5

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