EXERCISE - A
1) Write the first five terms of each of the following sequences whose nth terms are:
A) aₙ = 3n +2. 5,8,11,14,17
B) aₙ = (n -2)/3. -1/3,01/3,2/3,1
C) aₙ = 3ⁿ. 3,9,27,81,243
D) aₙ = (3n -2)/5. 1/5,4/5,7/5,2,13/5
E) aₙ = (-1)ⁿ. 2ⁿ. -2,4,-8,16,-32
F) aₙ = n(n -2)/2. -1/2,0,3/2,4,15/2
G) aₙ = n²-n +1. 1,3,7,13,21
H) aₙ = 2n² - 3n +1. 0,3,10,21,36
I) aₙ = (2n -3)/6. -1/6,6,1/2,5/6,7/6
2) Find the indicated terms in each of the following sequences whose nth terms are:
A) aₙ = 5n - 4.: a₁₂ and a₁₅. 56, 71
B) aₙ = (3n -2)/(4n+5). a₇, a₈. 19/33, 22/37
C) aₙ = n(n -1)(n -2). a₅, a₈ . 60,336
D) aₙ = (n -1)(2- n)(3+ n). a₁, a₂ , a₃. 0, 0,-12
E) aₙ = (-1)ⁿn. a₃, a₅, a₈. -3,-5,8
3) Find the next five terms of each of the following sequences given by:
A) a₁ , aₙ , aₙ₋₁ +2, n≥ 2. 3,5,7,9
B) a₁ = a₂ = 2 , aₙ= aₙ₋₁ - 3, n> 2. -1,-4,-7,-10,-13
C) a₁ =-1, a = (aₙ₋₁)/n, n≥ 2. -1/2, -1/6,-1/24,-1/120
D) a₁ = 4, aₙ = 4aₙ₋₁ +3, n >1. 19,79, 319, 1279
EXERCISE - B
1) Show that the sequence defined by aₙ = 5n -7 is an AP, find its common difference. 5
2) Show that the sequence defined by aₙ = 4n +5 is an AP, find its common difference. . 4
3) Show that the sequence defined by aₙ = 3n²- 5 is not an AP.
4) Show that the sequence defined by aₙ = 2n² + 1 is not an AP.
5) Show that the sequence defined by aₙ = -4n +15 is an AP, if yes, find its 15th term and common difference. -45,-4
6) The nth term of a sequence is 3n -2. Is the sequence an AP. ? If so , find its 10th term. 28
7) Prove that no matter what the real numbers a and b are, the sequence with nth term a+ nb is always an AP. What is the common difference? b
8) Write the sequence with nth term
A) aₙ = 3 + 4n. 7,11,15,19,....
B) aₙ = 5+ 2n. 7,9,11,13
C) aₙ = 6 - n. 5,4,3,2,1,0,-1,.....
D) aₙ = 9 - 5n. 4,-1,-6,-11,...
Show that all of the above sequences form AP.
EXERCISE - C
1) Find the common difference and write the next four terms of each of the following AP:
A) 1,-2,-5,-8,..... -3,-11,-14,-17,-20
B) 0,-3,-6,-9..... -3,-12,-15,-18,-21
C) -1,1/4,3/2,.... 5/4, 11/4, 16/4, 21/4, 26/4
D) -1,-5/6,-2/3,..... 1/6,-1/2,-1/3,-1/6,0
E) 51,59,67,75,..... 8, 83,91,99,107
F) 75,67,59,51,...... -8, 43,35,27,19
G) 1.8,2.0,2.2,2.4..... 0.2,2.6,2.8,3.0,3.2
H) 0, 4,1/2,3/4,..... 1/4, 1, 5/4
I) 119,136,153,170..... 17,187,204
J) 3,-2,-7,-12,........ -5,-17,-22,-27
2) Find out which of the following sequences are AP. For those which are AP, find out the common difference:
A) 3,6,12,24,..... No
B) 0,-4,-8,-12,........... Yes, -4
C) 1/2,1/4,6,1/8,........... No
D) 12,2,-8,-18,...... Yes, -10
E) 3,3,3,3,.............. Yes, 0
F) p, p+90, p+180, p+270,....... Where p= 999⁹⁹⁹. Yes, 90
G) 1,0,1.7,2.4,3.1,.... Yes, 0.7
H) -225,-425,-625,-825,.... Yes, -200
I) 10,10+2⁵,10+2⁶,10+2⁷,.... No
J) a+ b, (a+1)+ b, (a+1)+(b+1), (a+2)+(b+1), (a+2)+(b+2),......... Yes, 1
K) 1²,3²,5²,7²,............ No
L) 1²,5²,7²,73,....... Yes, 24
EXERCISE - D
1) Find:
A) 10th term of 1,4,7,10,..... 28
B) 18th term of √2,3√2,5√2,... 35√2
C) nth term of 13,8,3,-2,..... -5n+18
D) 10th term of -40,-15,10,35,... 185
E) 8th term of 117, 94, 71, 48,.... -44
F) 11th term of 10.0,10.5,11.0,11.5,.... 15
G) 9th term of 3/4,5/4,7/4,9/4... 19/4
H) 12th term of 8,13,18,23,28,.... 63
I) mth term of 8,13,18,23,28,.... 5m+3
J) 23rd term of 7,5,3,1,...... -37
K) nth term of 8,3,-2,-7,........ 13-5n
L) 7th term of 2,7,12,...... 32
M) 5th term of 21,28,35,............ 49
N) 18th term of 9,5,1,......... -59
2) Which term of the AP is
A) 3,8,13,.......is 248. 50
B) 84,80, 76,......is 0. 22
C) 4,9,14,........ is 254. 51
D) 21,42,63,84,....is 420. 20
E) -1,3,7,11,....is 95. 25
F) 4,9,14,19,......is 124. 25
G) 3,6,9,12,.......111. 37
H) 4,9,14,19,.....is 109. 22
I) 5,8,11,14,......is 320. 106
J) 64,60,56,..........is 0? 17
3) a) How many terms are there in the AP. 7,10,13,........43? 13
b) How many terms are there in the AP. -1, -5/6, -2/3, -1/2, ........10/3? 27
c) How many terms are there in the AP 7,13,19,.....,205? 34
d) How many terms are there in the AP 10,13,16,.....,43? 12
e) How many terms are there in the AP 5/6, ,1, 7/6,........10/3. 16
4)a) Is 68 a term of the AP 7,10,13,..? N
b) Is 302 a term of AP 3,8,13? N
c) Is 184 a term of the sequence 3,7,11,........? N
5) Find n if the given value of x is the nth term of the given AP:
A) 25,50,75,100,..,x= 1000. 40
B) -1,-3,-5,-7,....,x= -151. 76
C) 11/2, 11, 33/2, 22,....,x =550. 100
D)1, 21/11, 31/11, 41/11,...,x =171/11. 17
EXERCISE- E
1) Write the first six terms of an AP in which
A) a= 5, d= 4. 5,9,13,17,21,25
B) a= 98, d= -3. 98,95,92,89,86,83
C) a= 15/2, d= 3/2.
D) a= x, d= 3x+2. x, 4x+2,7x+4, 10x+6, 13x+8, 16x+10
2) Find a₃₀ - a₂₀ for the AP:
A) -9,-14,-19,-24,..... -50
B) a, a+ d, a+ 2d, a+ 3d, ...... 10d
3) Write the expression aₙ - aₖ for the AP:
a, a+ d, a+ 2d,......
Hence, find the common difference of the AP for which
A) 11th term is 5 and 13th term is 79. (n- k)d, 37
B) a₁₀ - a₅ = 200. 40
C) 20th term of 10 more than the 18th term. 5
4) Find the value of k if sequence is AP :
A) 8k+4, 6k -2, and 2k+7. 15/2
B) 2x, x+10, 3x +2 are in AP, find the value of x. 6
C) 8x+4, 6x-2, 2x+7. 15/2
D) x+1, 3x, 4x+2. 3
EXERCISE-F
1) The first term of an AP is -7 and the common difference 5. Find its 18th term and the general term. 78, 5n -12
2) The first term of an AP is 5, the common difference is 3 and the Last term is 80, find the number of terms
3) Write the 5th and 8th term of an AP whose 10th term is 43 and the common difference is 4. 23,35
4) The 6th and 17th terms of an AP are 19 and 41 respectively, find the 40th term. 87
5) The 10th and 18th terms of an AP are 41 and 73 respectively, find the 26th term. 105
6) The 6th and 8th terms of an AP are 12 and 22 respectively, find the second and nth term. -8, 5n -18
7) In an AP consists of 60 terms. If the first and the Last terms be 7 and 125 respectively, find 32nd term? 69
8) The first term of an AP is 5 and its 100th term is -292. Find the 50th term of this AP. -142
9) If 9th term of an AP is zero, Prove that its 29th term is double the 19th term.
10) If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that 25th term of the AP is zero.
11) In a certain AP the 24th term is twice the 10th term, show that 72nd term is twice the 34th term.
12) Find the arithmetic progression whose third term is 16 and seventh term exceeds it's fifth term by 12. 4,10,16,..
13) The seventh term of an AP is 32 and it's 13th term is 62. Find the AP. 2,7,12, 17,.......
14) The sum of 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 34. Find the first and the common difference. -1/2, 5/2
15) The 4th term of an AP is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference. -3,2
16) If (m+1)th term of an AP is twice the (n+1)th term, show that (3m+1)th term is twice the (m+n+1)th term.
17) Which term of an AP 3,10,17,....will be 84 more than its 13th term? 25th
18) If nth term of the AP 9,7,5,...is the same as the nth term of the AP 15,12,9,.... Find n. 7
19) Find the 12th term from the end of the following arithmetic progressions:
A) 3,5,7,9,.....201. 179
B) 3,8,13,.....,253. 198
C) 1,4,7,10,.....,88. 55
20) How many numbers of two digits are divisible by 3. 30
EXERCISE-G
1)a) The sum of three numbers in AP is -3, and their product is 8. Find the numbers. -4,-1,2 or 2,-1,-4
b) The sum of three numbers in AP is 21 and the product of the first and the third terms exceeds the second term by 6, Find the numbers. 1,7,13
c) Three numbers are in AP. If the sum of these numbers be 27 and the product 648, find the numbers. 6,9,12
d) The sum of three numbers in AP is 12 and sum of their cubes is 288. Find the numbers. 2,4,6 or 6,4,2
2)a) Find four numbers in AP whose sum is 20 and the sum of whose squares is 120. 2,4,6,8 or 8,6,4,2
b) Divide 32 into four parts which are in AP, such that the product of extremes is to the product of means is 7: 15. 2,6,10,14
c) Find four numbers in AP whose sum is 50 and in which the greatest number is 4 times the least. 5,10,15,20
d) The angles of a quadrilateral are in AP whose common difference is 10°. Find the angles. 75°,85°,95°105°
EXERCISE -H
1) Find the sum of the following arithmetic progressions:
a) 50,46,42,....to 50 terms. 320
b) 1,3,5,7,....to 12 terms. 144
c) 3,9/2,6,15/2,....to 25 terms. 525
d) 41,36,31,.....to 12 terms. 162
e) a+ b, a- b, a -3b,.....to 22 terms. 22a - 440b
f) (x-y)², x²+ y², (x+y)²,...to n terms. n[(x-y)²+ (n -1)xy]
g) (x-y)/(x+ y'), (3x- 2y/(x+y), (5x-3y)/(x+y),...to n terms. n{n(2x-y)-y}/2(x+y)
h) 1,4,7,10,......to 20 terms. 590
i) 5,2,-1,-4,-7,....to n terms. n(13-3n)/2
2) Find the sum of the first --
a) 11 terms of an AP: 2,6,10,14,... 242
b) 13 terms of an AP: -6,0,6,12,... 390
EXERCISE-I
1) Find n terms of an AP whose nth terms is given by
a) aₙ = 5 - 6n. n(2- 3n)
2) Find the sum of the first 15 terms of each of the following sequences having nth term as:
a) aₙ = 3+ 4n. 525
b) aₙ = 5+ 2n. 315
c) aₙ = 6 - n. 30
d) aₙ = 9 - 5n. -465
3) Find the sum of first 20 terms of the sequence whose nth is aₙ = B+ An. 210A+ 20B
4) Find the sum of first 25 terms of the sequence whose nth is aₙ = 2- 3n. -925
5) Find the sum of first 25 terms of the sequence whose nth is aₙ = 7 - 3n. -800
EXERCISE-J
1) Find the sum of first n natural numbers. n(n+1)/2
2) Find the sum of all odd numbers between 100 and 200. 7500
3) Find the sum of all integers between 84 and 719, which are multiples of 5. 50800
4) Find the sum of all integers between 50 and 500 which are divisible by 7. 17696
5) Find the sum of all even integers between 101 and 999. 246950
6) Find the sum of all integers between 100 and 550 which are divisible by 9. 16425
7) Find the sum of the first
a) 100 natural numbers. 5050
b) n natural numbers. n(n+1)/2
8) Find the sum of all three digit natural numbers, which are divisible by 7. 70336
9) Find the sum of all natural numbers between 250 and 1000 which are exactly divisible by 3. 156375
10) Find the sum of all integers between 2 and 100 which are divisible by 3. 867
11) Show that the sum of all integers between 1 and 1000 which are divisible by 3 is 83667.
EXERCISE-K
1) Find the sum:
a) 2+4+6+.......+200. 10100
b) 3+11+19+........+803. 40703
c) 1+3+5+7+........+109. 10000
d) (-5)+(-8)+(-11)+....+(-230). -8930
EXERCISE-L
1) How many terms of the series 54, 51, 48,....be taken so that their sum is 513? 18 or 19
2) How many terms of the series 20, 58/3, 56/3,....be taken so that their sum is 300 ? 36
3) How many terms of the series 18,16,14 ,....be taken so that their sum is zero? 19
4) How many terms are there in the AP whose first and fifth terms are -14, and 2 respectively and the sum of the terms is 40? 10
5) The first term of an AP is 2 and the Last term is 50. The sum of all terms is 442. Find the common difference. 3
6) If 12th term of an AP is -13 and the sum of the first four terms is 24, what is the sum of first 10 terms. 0
7) In an AP, if the first term is 22, the common difference is -4 and the sum to n terms is 64, find n. 4 or 8
8) In an AP, if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms? 1150
9) Find the sum of first 30 terms of an AP whose second term is 2 and seventh term is 22. 1680
10) Find the sum of first 20 terms of an AP in which third term is 7 and seventh term is 2 more than thrice of its 3rd term. 740
11) If the sum of a certain number of terms starting from first of n AP is 25,22,19,... is 116. Find the last term. 4
12) The third term of an AP is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the the sum of first 20 terms. -1,4,740
13) If the mth term of an AP is 1/n and the nth term is 1/m, show that the sum of mn terms is (mn+1)/2.
14) If in an AP the sum of m terms is equal to n and the sum of n terms is equal to m, then show that the sum of (m+ n) terms is -(m+ n).
15) If in an AP the sum of m terms is equal to the sum of n terms, then show that the sum of (m+ n) terms is 0.
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