Arithmetic progressions - Problem based video on Arithmetic progressions part 4 for class 10th maths.
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2. Q) Which term of the sequence 23, 22 , 22, 21 … is
the first negative term ?
Given : Sequence = 23, 22
ퟏ
ퟐ
, 22, 21
ퟏ
ퟐ
1
2
1
2
Problems based on
Arithmetic Progressions
To find: First negative term
Chapter : Arithmetic Progressions Website: www.letstute.com
3. Problems based on
Arithmetic Progressions
1
2
Solution: a = first term = 23, 2nd term = 22 and
d = common difference = 22 - 23 = - 1
2
1
2
Then, an < 0
a + (n – 1)d < 0
23 + (n – 1)
−ퟏ
ퟐ
< 0
Chapter : Arithmetic Progressions Website: www.letstute.com
4. n
2
Problems based on
Arithmetic Progressions
1
2
23 - + < 0
47 - < 0
2
n
2
47 <
2
n
2
47 < n
n > 47
Since, 48th is the natural number just greater than 47, therefore
n = 48.
Thus, 48th term of the given sequence is the first negative term.
Chapter : Arithmetic Progressions Website: www.letstute.com
5. Problems based on
Arithmetic Progressions
1
n
1
m
Q) If the mth term of an AP be and nth term be , then
Show that its (mn)th term is 1.
Given: mth term =
ퟏ
퐧
nth term =
ퟏ
퐦
To prove: (mn)th term is 1
Chapter : Arithmetic Progressions Website: www.letstute.com
6. Problems based on
Arithmetic Progressions
Solution: Let a = first term and d = common difference of the
given AP.
am = a + (m – 1)d
= a + (m – 1)d ∴ am 1 = , given … (1)
n
1
n
and an = a + (n – 1)d
1
m
1
m
= a + (n – 1)d ∴ an = , given … (2)
Chapter : Arithmetic Progressions Website: www.letstute.com
7. Problems based on
Arithmetic Progressions
Subtracting equation (2) from equation (1), we get
1
n
1
m
- = [a + (m – 1)d] - [a + (n – 1)d]
1
n
-
1
m = a + (m – 1)d - a – (n – 1)d
1
n
- 1
m
= (m – 1– n + 1)d
m - n
mn
= (m– n)d
d = 1 … (3)
mn
Chapter : Arithmetic Progressions Website: www.letstute.com
8. Substituting the value of ‘d’ in equation (2), we get,
1
m = a + (n – 1) 1
mn
1
m
- (n – 1)
mn
= a
a = n – n + 1
mn
a = 1
mn
….(4)
Problems based on
Arithmetic Progressions
Chapter : Arithmetic Progressions Website: www.letstute.com
9. Problems based on
Arithmetic Progressions
(mn)th term = amn = a + (mn – 1)d
amn = 1
mn
1
mn
+ (mn – 1) [Using (3) and (4)]
amn = 1
mn
mn - 1
mn
+
amn = 1+ mn - 1
mn
mn
mn
= = 1
Hence, the (mn)th term is 1
Chapter : Arithmetic Progressions Website: www.letstute.com
11. Now we know…
Problems based on
Arithmetic Progressions
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Chapter : Arithmetic Progressions Website: www.letstute.com
12. Next video….
Some more problems based on
Arithmetic Progressions
Part 4
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Chapter : Arithmetic Progressions Website: www.letstute.com
13. Problems based on
Arithmetic Progressions
Q) The sum of three consecutive numbers in AP is - 6,
and their product is + 64. Find the numbers.
Given: S3 = - 6
Product of three consecutive numbers = 64
To Find: 3 consecutive numbers
Chapter : Arithmetic Progressions Website: www.letstute.com
14. Problems based on
Arithmetic Progressions
Solution: Let the numbers be (a – d), a, (a + d)
Sum = - 6
(a – d) + a + (a + d) = - 6
3a = - 6
a = - 2 …(1)
Chapter : Arithmetic Progressions Website: www.letstute.com
16. Problems based on
Arithmetic Progressions
If d = + 6, the numbers are (–2 – 6), – 2 and (– 2 + 6),
i.e, – 8, –2 and 4
If d = – 6, the numbers are [– 2 – (– 6)], – 2 and [– 2 + (– 6)],
i.e, 4, –2 and – 8
Hence, the required numbers are – 8, – 2, 4 or 4, – 2, – 8.
Chapter : Arithmetic Progressions Website: www.letstute.com
17. Problems based on
Arithmetic Progressions
Q) Find the four consecutive even number of terms in AP
whose sum is 16 and the sum of whose squares is 84.
Given: S4 = 16
Sum of squares of 4 consecutive even number of terms = 84
To Find: 4 consecutive even number of terms in AP
Chapter : Arithmetic Progressions Website: www.letstute.com
18. Problems based on
Arithmetic Progressions
Solution: Let the four consecutive even number of terms in AP
be a – 3d, a – d, a + d, a + 3d
Sum of the four consecutive even number of terms = 16
(a – 3d) + (a - d) + (a + d) + (a + 3d) = 16
4a = 16
a = 4 …(1)
Chapter : Arithmetic Progressions Website: www.letstute.com
19. Problems based on
Arithmetic Progressions
Sum of the squares of the four consecutive even number of
terms in AP = 84
(a – 3d)2 + (a - d)2 + (a + d)2 + (a + 3d)2 = 84
a2 - 6ad + 9d2 + a2 - 2ad + d2 + a2 + 2ad + d2 + a2 +6ad + 9d2 = 84
4a2 + 20d2 = 84
4(a2 + 5d2) = 84
a2 + 5d2 = 21
42 + 5d2 = 21 [Using (1)]
5d2 = 5
d2 = 1
d = + 1
Chapter : Arithmetic Progressions Website: www.letstute.com
20. Problems based on
Arithmetic Progressions
If d = + 1 then the terms are (4 – 3), (4 – 1), (4 + 1), (4 + 3)
i.e. 1, 3, 5, 7.
If d = - 1 then the terms are (4 + 3), (4 + 1), (4 - 1), (4 - 3)
i.e. 7, 5, 3, 1.
Hence, the required consecutive even number of terms are
1, 3, 5, 7 or 7, 5, 3, 1.
Chapter : Arithmetic Progressions Website: www.letstute.com
21. Problems based on
Arithmetic Progressions
nth term from the end
Let the AP be a, a + d, a + 2d, ….. l
First term = a, common difference = d and last term = l
The AP may be written as
a, (a + d), (a + 2d),…. (l - 2d), (l – d), l
Last term from the end is l = l – (1 – 1)d
Second term from the end is l – d = l – (2 – 1)d
Third term from the end is l – 2d = l – (3 – 1)d
Fourth term from the end is l – 3d = l – (4 – 1)d … and so on.
The nth term from the end is l – (n – 1)d
Chapter : Arithmetic Progressions Website: www.letstute.com
22. Problems based on
Arithmetic Progressions
Q)Find the 12th term from the end of the AP 3, 6, 9,…60
Given: AP = 3, 6, 9,…60
To Find: 12th term from the end
Chapter : Arithmetic Progressions Website: www.letstute.com
23. Problems based on
Arithmetic Progressions
Solution: d = common difference = 6 – 3 = 3 and
the last term = l = 60
nth term from the end = l – (n – 1)d
∴ 12th term from the end = 60 – (12 – 1) x 3
= 60 – 11 x 3
= 60 – 33 = 27
Hence, the 12th term from the end of the AP 3, 6, 9… 60 is 27
Chapter : Arithmetic Progressions Website: www.letstute.com
24. Problems based on
Arithmetic Progressions
Treasure
Finding out exact numbers when the
sum and product of numbers is given.
nth term from the end using the formula l – (n – 1)d
Chapter : Arithmetic Progressions Website: www.letstute.com
25. Now we know…
Problems based on
Arithmetic Progressions
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Chapter : Arithmetic Progressions Website: www.letstute.com