ArithmeticProgression

35
UNIT-5
ARITHMETIC PROGRESSIONS
One of the endlessly alluring aspects of mathematics is that its
thorniest paradoxes have a way of blooming into beautiful theories
1. The fourth term of an AP is 0. Prove that its 25th
term is triple its 11th
term.
Ans: a4= 0
⇒ a + 3d = 0
T.P a25= 3 (a11)
⇒ a + 24d = 3 (a + 10d)
⇒ a + 24d = 3a + 30d
RHS sub a = - 3d
- 3d + 24d = 21d
LHS 3a + 30d
- 9d + 30d = 21d
LHS = RHS
Hence proved
2. Find the 20th
term from the end of the AP 3, 8, 13……..253.
Ans: 3, 8, 13 ………….. 253
Last term = 253
a20 from end
= l – (n-1)d
253 – ( 20-1) 5
253 – 95
= 158
3. If the pth
, qth
& rth
term of an AP is x, y and z respectively,
show that x(q-r) + y(r-p) + z(p-q) = 0
Ans: pth
term ⇒ x = A + (p-1) D
qth
term ⇒ y = A + (q-1) D
rth
term ⇒ z = A + (r-1) D
T.P x(q-r) + y(r-p) + z(p-q) = 0
={A+(p-1)D}(q-r) + {A + (q-1)D} (r-p)
+ {A+(r-1)D} (p-q)
A {(q-r) + (r-p) + (p-q)} + D {(p-1)(q-r)
+ (r-1) (r-p) + (r-1) (p-q)}
⇒ A.0 + D{p(q-r) + q(r-p) + r (p-q)
- (q-r) – (r-p)-(p-q)}
= A.0 + D.0 = 0.

36
Hence proved
4. Find the sum of first 40 positive integers divisible by 6 also find the sum of first
20 positive integers divisible by 5 or 6.
Ans: No’s which are divisible by 6 are
6, 12 ……………. 240.
S40 = [ ]2406
2
40
+
= 20 x 246
= 4920
No’s div by 5 or 6
30, 60 …………. 600
[ ]60030
2
20
+ = 10 x 630
= 6300
5. A man arranges to pay a debt of Rs.3600 in 40 monthly instalments which are in a
AP. When 30 instalments are paid he dies leaving one third of the debt unpaid.
Find the value of the first instalment.
Ans: Let the value of I instalment be x S40 = 3600.
⇒ [ ]da 392
2
40
+ =3600
⇒2a + 39d = 180 - 1
S30 = [ ]da 292
2
30
+ =2400
⇒30a + 435d = 2400
⇒2a + 29d = 160 - 2
Solve 1 & 2 to get
d = 2 a = 51.
∴ I instalment = Rs.51.
6. Find the sum of all 3 digit numbers which leave remainder 3 when divided by 5.
Ans: 103, 108……….998
a + (n-1)d = 998
⇒ 103 + (n-1)5 = 998
⇒ n = 180
S180 = [ ]998103
2
180
+
= 90 x 1101
S180 = 99090

37
7. Find the value of x if 2x + 1, x2
+ x +1, 3x2
- 3x +3 are consecutive terms of an
AP.
Ans: a2 –a1
=
a3 –a2
⇒ x2
+ x + 1-2x - 1 = 3x2
– 3x + 3- x2
-x-1
x2
- x = 2x2
– 4x + 2
⇒ x2
- 3x + 2 = 0
⇒ (x -1) (x – 2) = 0
⇒ x = 1 or x = 2
8. Raghav buys a shop for Rs.1,20,000.He pays half the balance of the amount in
cash and agrees to pay the balance in 12 annual instalments of Rs.5000 each. If
the rate of interest is 12% and he pays with the instalment the interest due for the
unpaid amount. Find the total cost of the shop.
Ans: Balance = Rs.60,000 in 12 instalment of Rs.5000 each.
Amount of I instalment = 5000 +
100
12
60,000
II instalment = 5000 + (Interest on unpaid amount)
= 5000 + 6600 



55000x
100
12
= 11600
III instalment = 5000 + (Interest on unpaid amount of Rs.50,000)
∴ AP is 12200, 11600, 11000
D = is 600
Cost of shop = 60000 + [sum of 12 instalment]
= 60,000 +
2
12
[24,400-6600]
= 1,66,800
9. Prove that am + n + am - n =2am
Ans: a m + n = a1 + (m + n - 1) d
a m-n = a1 + (m - n -1) d
am = a1 + (m-1) d
Add 1 & 2
a m+n + a m-n = a1+(m+n-1) d+ a1 + (m-n-1)d
= 2a1+(m+n+m-n-1-1)d
= 2a1+ 2(m-1)d

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= 2[a1+ (m-1)d]
= 2[a1+ (m-1)d]
= 2am. Hence proved.
10. If the roots of the equation (b-c)x2
+(c-a)x +(a-b) = 0 are equal show that a, b, c
are in AP.
Ans: Refer sum No.12 of Q.E.
If (b-c)x2
+ (c-a)x + (a-b)x have equal root.
B2
-4AC=0.
Proceed as in sum No.13 of Q.E to get c + a = 2b
⇒ b - a = c - b
⇒ a, b, c are in AP
11. Balls are arranged in rows to form an equilateral triangle .The first row consists of
one ball, the second two balls and so on. If 669 more balls are added, then all the
balls can be arranged in the shape of a square and each of its sides then contains 8
balls less than each side of the triangle. find the initial number of balls.
Ans: Let their be n balls in each side of the triangle
∴ No. of ball (in ∆) = 1 + 2+ 3………..=
( )
2
1+nn
No. of balls in each side square = n-8
No. of balls in square = (n-8)2
APQ
( )
2
1+nn
+ 660 = (n-8)2
On solving
n2
+ n + 1320 = 2(n2
- 16n + 64)
n2
- 33n - 1210 = 0
⇒ (n-55) (n+22) = 0
n=-22 (N.P)
n=55
∴No. of balls =
( )
2
1+nn
=
2
56x55
= 1540
12. Find the sum of )
3
1()
2
1()
1
1(
nnn
−+−+− ……. upto n terms.
Ans: 





−+





−
nn
2
1
1
1 - upto n terms
⇒[1+1+…….+n terms] – [
n
1
+
n
2
+….+ n terms]
n –[Sn up to n terms]

39
Sn =
2
n
[2a + (n-1)d] (d =
n
1
, a =
n
1
)
=
2
n




−+
n
n
n
1
)1(
2
=
2
1+n
(on simplifying)
n -
2
1+n
=
=
2
1−n
Ans
13. If the following terms form a AP. Find the common difference & write the next
3 terms3, 3+ √2, 3+2√2, 3+3√2……….
Ans: d= 2 next three terms 3 + 4 2
, 3 + 5 2
, 3 + 6 2
……..
14. Find the sum of a+b, a-b, a-3b, …… to 22 terms.
Ans: a + b, a – b, a – 3b, up to 22 terms
d= a – b – a – b = 2b
S22 =
2
22
[2(a+b)+21(-2b)]
11[2a + 2b – 42b]
= 22a – 440b Ans.
15. Write the next two terms √12, √27, √48, √75……………….
Ans: next two terms 108 , 147 AP is 2 3 , 3 3 , 4 3 , 5 3 , 6 3 , 7 3 ……
16. If the pth
term of an AP is q and the qth
term is p. P.T its nth
term is (p+q-n).
Ans: APQ
ap = q
aq = p
an = ?
a + (p-1) d = q
a + (q-1) d = p
d[p – q] = q – p Sub d = -1 to get ⇒ = -1 ⇒ a = q + p -1
an = a + (n – 1)d
= a + (n - 1)d
= (q + p – 1) + (n – 1) - 1
an = (q + p – n)

40
17. If
5
1
,
3
1
,
2
1
+++ xxx
are in AP find x.
Ans:
5
1
,
3
1
,
2
1
+++ xxx
are in AP find x.
3
1
5
1
2
1
3
1
+
−
+
=
+
−
+ xxxx
⇒
158
2
65
1
22
++
=
++ xxxx
On solving we get x = 1
18. Find the middle term of the AP 1, 8, 15….505.
Ans: Middle terms
a + (n-1)d = 505
a + (n-1)7 = 505
n – 1 =
7
504
n = 73
∴ 37th
term is middle term
a37 = a + 36d
= 1 + 36(7)
= 1 + 252
= 253
19. Find the common difference of an AP whose first term is 100 and sum of whose
first 6 terms is 5 times the sum of next 6 terms.
Ans: a = 100
APQ a1 + a2 + ……. a6 =5 (a7 + …….. + a12)





 +
2
6
61 aa
=5 x 6 




 +
2
127 aa
⇒ a + a + 5d = 5[a + 6d + a + 11d]
⇒ 8a + 80d = 0 (a = 100)
⇒ d = - 10.
20. Find the sum of all natural no. between 101 & 304 which are divisible by 3 or 5.
Find their sum.
Ans: No let 101 and 304, which are divisible by 3.
102, 105………….303 (68 terms)
No. which are divisible by 5 are 105, 110……300 (40 terms)

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No. which are divisible by 15 (3 & 5) 105, 120…… (14 terms)
∴ There are 94 terms between 101 & 304 divisible by 3 or 5. (68 + 40 – 14)
∴ S68 + S40 – S14
= 19035
21. The ratio of the sum of first n terms of two AP’s is 7n+1:4n+27. Find the ratio
of their 11th
terms .
Ans: Let a1, a2… and d1, d2 be the I terms are Cd’s of two AP’s.
Sn of one AP =
274
17
+
+
n
n
Sn of II AP
[ ]
[ ]
=
−+
−+
22
11
)1(2
2
)1(2
2
dna
m
dna
m
274
17
+
+
n
n
⇒ =
−+
−+
22
11
)1(2
)1(2
dna
dna
274
17
+
+
n
n
We have sub. n = 21.
=
+
+
22
11
202
202
da
da
27)21(4
1217
+
+x
⇒
111
148
10
10
22
11
=
+
+
da
da
=
3
4
∴ ratio of their 11th
terms = 4 :3.
22. If there are (2n+1)terms in an AP ,prove that the ratio of the sum of odd terms
and the sum of even terms is (n+1):n
Ans: Let a, d be the I term & Cd of the AP.
∴ ak = a + (k – 1) d
s1 = sum to odd terms
s1 = a1 + a3 + ……… a 2n + 1
s1 = [ ]1n21 aa
2
1n
++
+
= [ ]nd2a2
2
1n
1 +
+
s1 = (n + 1) (a + nd)
s2 = sum to even terms
s2 = a2 + a4 + ….. a 2n

42
s2 = [ ]n22 aa
2
n
+
=
2
n
[a + d + a + (2n – 1)d]
=n [a + nd]
∴ s1 : s2 =
)(
))(1(
ndan
ndan
+
++
=
n
n 1+
23. Find the sum of all natural numbers amongst first one thousand numbers which
are neither divisible 2 or by 5
Ans: Sum of all natural numbers in first 1000 integers which are not divisible by 2 i.e.
sum of odd integers.
1 + 3 + 5 + ………. + 999
n = 500
S500 =
2
500
[1 + 999]
= 2,50,000
No’s which are divisible by 5
5 + 15 + 25 …….. + 995
n = 100
Sn =
2
100
[5 + 995]
= 50 x 1000 = 50000
∴ Required sum = 250000 – 50,000
= 200000

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