---------------- ****** -------------------
EXERCISE - A
1) Write first 4 terms in each of the sequences:
A) aₙ = (5n+2). 7, 12, 17, 22
B) aₙ = =(2n-3)/4. -1/4,1/4,3/45/4
C) aₙ = =(-1)ⁿ⁻¹ x 2ⁿ⁺¹. 4,-8,16,-32
2) find first five terms of the sequence defined by
a₁= 1, aₙ= aₙ₋₁+3 for n≥2. 1, 4, 7, 10, 13
3) find first five terms in the sequence, defined by
a₁= -1, aₙ= (aₙ₋₁)/n for n≥2. -1, -1/2,-1/6,-1/24,-1/120
EXERCISE - B
1) FIND
a) 2, 6, 10, ...... 20th term. 78
b) 1, 3, 5 ..... 100th term. 199
c) 23rd term of 7,5,3,1,.... -37
d) 20th term √2, 3√2, 5√2,... 39√2
e) nth term 8, 3, -2,-7,....... 13-5n
f) (a-3b), (a- b),(a+b) ... r th term. a+ 2br - 5b
g) 13, 8, 3,-2.... nth term. 18-5n
h) 3,5,7,9,... 8th and nth term. 17, 2n+1
i) 2, 3/2,2,.....nth & 10th term. (n+3)/2, 13/2
j) 1, (n+2)/n, (n+4)/n, (n+6)/n, .... 12th and n th term. (n+22)/n, (3n-2)/n
EXERCISE - C
1) Which term of the sequences
a) 5, 2, -1, ......is -22. 10
b) 148, 146, 144 ..... is 30. 60
c) 5, 8,11, 14,.... is 320 ? 106th
d) 64, 60, 56,...is 0 ? 17th
e) 10,13,16, ...., 43 ? 12
f) 5/6, 1, 7/6 ,.... 10/3 ? 16
g) 4,9,14,19,.... is 109? 22
h) 7,13,19,...., 205. 34
i) 84,80,76,....... Is 0 . 22
j) 2,5,8..... is 92. 31
k) -1,-5/6,-2/3,-1/2,...10/3. ? 27
l) 12+i,11+6i,10+4i,. ...is (i) purely real (ii) purely imaginary.
EXERCISE - D
1) a) Is 305 a term of the series 7, 10, 13, 16? If not find nearest term. No,101th
b) Is 90 a term of the series 4,7, 10, 13 .... ? B. No
c) Is 68 a term of A.P 7,10,13,.. no
d) Is 302 a term of 3,8,13.....? No
e) Is 301 a term of the AP 5, 11, 17, 23.....? No
EXERCISE - E
1) Find the value of k if series in AP
a) 2k-1,k+1,3-k. 0
b) 5k-1, 3k+7,8k+1. 2
c)k²+2k+2,3k²+6k+6,4k²+5k+4. -3,-2
EXERCISE - F
1) Find
a) 15th term from the end of the AP 3,5,7,9,... 201. 173
b) 19th term from the end of the AP 2,6,10,14,...86.
c) 15th term from the end of 7,10,13, ....130. 88
d) 20th term from the end of 3,8,13, ..... 253. 158
e) 12 th term from the end of 3,5,7,9,......201. 179
f) 12 th term from the end of 1,4,7,10,....,88. 55
EXERCISE - G
1) Find 10th term of the AP where 1st term is 5 & common difference is 2. 23
2) 1st term of an AP is 6, common difference is 2. find 15th term.
3) The 20th term of an A. P is 79. If 1st term is 3, find 10th term. 39
4) The 5th and 13th terms of an AP are 5 and - 3 respectively. find this AP and obtain its 16th term. 9, 8, 7, 6 and 16th term=-6
5) Find 15th term of an AP whose 10th term is - 25 & 20th term is - 55. -40
6) Find 2nd term and rth term of the A. P whose 6th term is 12 and 8th term is 22. -5, 5r - 18
7) The 6th &16th term of AP are 9√2/2 and 19√2/2. Find its 20th term. 23/√2
8) The 2nd, 31st and the last terms of an AP are 31/4,1/2 and -13/2 respectively. find 1st term and the number of terms. 8, 59
9) If the 9th term of an AP is 0, prove that its 29th term is double the 19th term.
10) how many two digit numbers are divisible by 7 ? 13
11) The 4th terms 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
12) if 7 times the 7th term of an AP is equals to 11 times its 11th term, show that its 18th term is 0.
13) If 9th term of an A. P. Is zero, prove that it's 29th term is double the 19th term.
14) If 10th times the 10th term of an A. P is equal to 15 times 15th term, show that 25th term of an A. P is zero.
15) How many Numbers of two digitare Divisible by 3.
16) The first term of an AP is 5, 11th term is 125. Find common difference.
17) The 8th term of an AP is double the 13th. Is 2nd term is double the 10th term.
18) The 4th and 8th term is 2 and 10. Find 11th term and first term.
19) A certain series in AP. Every
term of it is multiplied by 3. Is resultant series in AP ?
20) a) In 3rd term: 5th term=1: 4 Find 7th term: 12th term.
b) If 7 times the 7th term is equal to 11 times of the 11th term. Find the 18th term ?
21) The 4th term is equal to 3 times the first term and the 7th term exceeds twice the third term by 1. Find 1st term and common difference.
22) if the mth term of an AP be 1/n and its nth term be 1/m then show show that its mn th term is (1+m-n)/mn
23) If the mth term of a given AP is n and its nth term is m then show that its pth term is n+ m -p
24) If m times the mth term of an AP is equal to n times its nth term, show that m + n th term of the AP is zero.
25) If the pth, qth and rth term of an AP be a, b, c respectively. show that (q-r)a + (r-p)b +(p-q)c = 0
26) Show that the nth term from the end of an AP with the first term a, common difference d and the last term l is given by (a+l+ad-and)/a
27) If x th term of AP is y and y th term is x. Find (x+y) th term and m th term.
28) if (m+1) th term of an A. P. is twice the (n+1)th term, prove that (3m+1)th term is twice the (m+n+1)th term.
29) The 4th term of an A. P is is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
EXERCISE - H
1) Which term of the AP
a) 19, 91/5, 92/5, ........is the negative term ? 28
b) Which term of the sequence 20, 77/4, 37/2,71/4 is the 1st negative term ? 28
c) Which term of the sequence 24,93/4,45/2,87/4 is the first negative term ? 31
EXERCISE - I
1) Find A. M between
a) 9 and 19. 14
b) 12 and -8. 2
c) -13 and -7. -10
d) 14 and 18. 16
e) (a-b) and (a+b). a
2) Fill up the gap
a) 4 , _ , _ , 10
b) _ , _ , _ , -2 , -10.
c) 34, _ , _ , _ , 48.
3) Insert 3 AM between
a) 3 and 19. 7,11,15
b) 2 and 14
c) - 24 and 0
d) 13/4 and 5/4
4) Insert 5 arithmetic means between
a) 8 and 26. 11,14,17,20,13
b)17 and 29.
c) 2 and 26
5) Insert 4 A. M between
a) 4 and 29. 9,14,19,24
b) (a - b)² and a² + b² + 8ab.
c) 4 and 19. 7,10,13,16
6) Insert 7 A. M between√2 and 17√2.
7) Find the middle term of
a) 1,6,11,......101. 11th, 51
b) 4,7,10, ....73. 12th,13th, 24
c) 5,8,11, ...., 95.
EXERCISE - J
1)a) Find three numbers in AP whose sum is 24 and product of extreme is 55 5,8,11 or 11,8,5
b) three numbers are in AP. If their sum is 27 and the product 648, find the numbers. 6,9,12
c) the sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. find these terms. 4,7,10
d) The three Numbers are in 3:7:9 if 5 is substracted from the second the resulting Number are in AP. Find the original number.
e) The sum of three terms of an A. P. is 21 and the product of the first and the third term exceeds the second term by 6, find three nos.
f) The sum of three Numbers in AP is 12 and sum of their cubes is 408. Find the numbers.
g) Divide 12 into 3 parts in AP as
their squares is 56.
2a) find the 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 4 parts which are in AP such that the product of the extremes is to product of means as 7:15. 2,6,10,14
c) the angles of a quadrilateral are in AP whose common difference is 10°. find the angles. 75,85,95,105
d) Find the four numbers in A. P., Whose sum is 50 and in which the greatest number is 4 times the least.
e) The sum of four numbers in AP is 24. Sum of their squares is 164. Find the numbers.
f) Divide 20 into four parts in AP
and such that the product of first and fourth is to the product of second and third in the ratio 2:3.
g) Divide 32 into four parts which
are in AP. Such that the product of extremes is to the product of mean is 7:15.
EXERCISE - K
1) Find the sum of the series
a) 1,3, 5, 7... 24th term. 576
b) 2+5+8+..... 30th term
c) 6, 16/3, 14/3... 10th term. 30
d) 0.8+1.2+1.6+... to 25th terms.
e) 2+ 7 +12+ .... upto 20 terms.
f) √2,2√2,3√2...20th term. 210√2
g) 0.7,0.71,0.72...100th term. 119.5
h) (n-1)/2 + n/n + (n+1)/n + .... upto n terms.
i) 101+99+97+....+47. 2072
2) How many term
a) 26+21+16+11.....to get 11. 11
b) 18+16+14+12....to get 78. Explain the double answer. 6,13
EXERCISE - L
1) Find the series
a) 3p²+5p.
b) an²+bn.
c) 2n+ 3n²
d) 2p² + p
e) The sum of n terms of an AP is 3n²+5n. Find p th term. Find which term of the A. P is 152.
2) Find the series if
a) 4r - 3.
b) 10 - 3n
EXERCISE - M
1) The first and the last terms of an A. P having finite number of terms are respectively - 2 and 124 and the sum of the A. P is 6161. Find the number of terms.
2) The first and the last term of an A. P are respectively - 4 and 146, and the sum of A. P= 7171. Find the number of terms, common difference.
3) If the sum of first m terms of an AP is equal to the sum of the first n terms, prove that the sum of the first (m+n)th term is zero.
4) In an AP if the 12th term is -13
and sum of first four terms is 24. What is the sum of its first 10 terms.
5) If the first term of an AP is 2 and sum of the first five terms is equal to one-fourth of the sum of the next five terms. Find the sum of first 30 terms. Also find its 20th term.
6) If the sum of p terms of an AP is q and the sum of q terms is p. Show that the sum of p+q terms is -(p+q). Also find the sum of first p-q terms.
7) How many terms of an AP -6, -11/2, -5,... make the sum -25 ?
8) How many terms of the sequence 18,16,14, ... Should be taken that sum is zero.
EXERCISE - N
1)a) Find the sum of all Numbers between 100 to 300 which are Divisible by 8.
b) Find the sum of all integer from 200 to 400 not Divisible by 7.
c) Find the sum of all odd numbers of four digits which are Divisible by 9.
d) Find the sum of all integers between 50 and 500 which are Divisible by 7 ?
e) Find the sum all natural
numbers between 100 and 1000 (excluding both) which are Divisible by 5 ?
f) The first, second and the last terms of an AP are a, b, 2a. Find its sum.
MISCELLANEOUS (1)
1) The sum of the first p,q,r terms of an A.P are a,b,c respectively. Show that (aq-ar)/p + (br-bp)/q + (cp-cq)/r =0
2) The sums of n terms of three arithmetical progression S' , S" S"'. The first term of each is unity and the common difference are 1,2,3 respectively. Prove S'+S"'= 2 S".
3) The pth term of an A. P. is a and qth term is b. Prove the sum of its p+q is (p+q)/2 {a + b + (a-b)/(p-q)}
4) The sum of n terms of two arithmetic progression are in the ratio (3n+8):(7n+5). Find the ratio of their 12th terms.
5) The interior angles of a polygon in A. P. The smallest angle is 120 and the common difference is 5°. Find the number of sides of the polygon.
48) If (aⁿ + bⁿ)/(aⁿ⁻¹+ bⁿ⁻¹) is the AM between a and b then find the value of n. 1
6) There are n AM between 5 and 23. Such that the ratio of 1st mean : last mean is 1:3. Find the value of n.
7) 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 show that (p+q)= (r+s).
8) 7th term : 3rd term = 12 : 5 Find 13th term : 4th term.
9) If nth term of the A. P. 9,7,5, is same as the nth term of the A. P 15,12,9,.... Find n.
EXERCISE - O
1) If a,b,c are in A. P prove the following are in A. P.
a) b+c, c+a, a+b.
b) a²(b+c),b²(c+a), c²(a+b).
c)a(1/a+1/c),b(1/c+1/a),c(1/a+1/b)
d) {(b+c)²-a²},{(c+a)²-b²},{(a-b)²-c²}.
e) (a- c)² = 4(b² - ac).
f) a³ + 4b³+ c³= 3b(a²+c²).
g) a²(b+c), b²(c+a), c²(a+b) .
h) b+c -a, c+a - a, a+b - c.
i) b c - a², ca - b² , ab - c².
j) (a - c)² = 4(a - b)(b - c).
k) a² + c² +4ac = 2(a b+ b c+ c a).
l) a³+c³+6abc = 8b³.
2) If a², b²,c² are in A. P then prove that the following are in also in A. P.
3) 1/(b+c), 1/(c+a), 1/(a+b).
b) a/(b+c), b/(c+a), c/(a+b)
c) a/(b+c),b/(c+a), c/(a+b).
4) If a²(b+c), b²(c+a), c²(a+b) are in A. P prove that either a,b,c are in A. P or ab+b c+ca=0.
5) 1/a, 1/b, 1/c are in A. P prove following are in A. P.
a) (b+c)/a, (c+a)/b, (a+b)/c
b) a(b+c), b(c+a), c(a+b).
6) (b+c)/a, (c+a)/b, (a+b)/c are in A. P prove following are in A. P.
a) 1/a, 1/b, 1/c.
b) b c, c a, a b
EXERCISE -P
1) Find the sum of
a) 3.7+5.10+7.13+...... n terms.
b) 7²+8²+9²+ ...20².
c) 10²+11²+12²+......30².
d) 6³+7³+8³+.....20³.
e) 1.2²+3.5²+5.8²+.....n terms.
MIXED PROBLEMS
No comments:
Post a Comment