Arithmetic progressions - Poblem based Arithmetic progressions

Chapter : Arithmetic Progressions Website: www.letstute.com

Arithmetic Progressions

Problems based on Arithmetic Progressions

Q) Show that -3, 0, 3, 6, 9, …. is an AP. Find its 25th term and the general term.

Solution: We have 0 - (-3) = 3 - 0 = 6 - 3 = 9 - 6 = 3, which is a constant. Therefore, the given sequence is an AP with a common difference = 3.

a = first term = - 3 and d = common difference = 3

an = a + (n - 1)d

a25 = - 3 + (25 - 1) (3)



a25 = -3 + (24) (3)

a25 = -3 + (72) = 69

an = a + (n - 1) dGeneral term,an = -3 + (n - 1) 3

an = -3 + 3n - 3

an = 3n - 6

Hence, a25 = 69 and an = 3n - 6



Q) Show that the sequence defined by an = 2n - 1, is an AP. Find its 11th term.

Solution: an = 2n - 1

Replacing n by n - 1 we get, an – 1 = 2(n - 1 ) - 1

Now, an - an – 1 = 2n - 1 - [2(n - 1) - 1]

= 2n - 1 - 2n + 2 + 1 = 2

Thus, the given sequence is an AP with a constant difference 2.



an = 2n - 1

a11 = 2 x 11 - 1

a11 = 22 - 1 = 21



Hence, a11 = 21

Q) How many terms are there in the AP 2, 9, 16, … 261 ?

Solution:

an = a + (n – 1) d

Thus, the given AP has 38 terms.

a = first term = 2 and d = common difference = 9 – 2 = 7

Suppose there are n terms in the given AP, then nth term = 261

261 = 2 + (n – 1) 7 261 = 2 + 7n – 7

7n = 266

n = 38



Q) Find the value of the middle term (s) of the AP -11, -7, -3, ….49

Solution:

an = a + (n - 1) d

a = first term = -11, an = 49

49 = -11 + (n - 1) x 4

60 = 4n - 4

64 = 4n

n = 16

d = common difference = -7 (-11) = 4

Now



As n is an even number, there will be two middle terms andth

i.e 8th term and the 9th term

a8 = a + 7d = -11 + (7 x 4) = 17

a9 = a + 8d = -11 + (8 x 4) = 21

Hence, the values of two middle terms are 17 and 21, respectively.



(𝟏𝟔𝟐 +𝟏)

Q) Is 63 a term of the AP -1, 4, 9, 14,……?

Solution: a = first term = -1 and d = common difference = 4 - (-1) = 4 + 1 = 5

Let the nth term of the given AP be 63

Then an = 63

a + (n - 1) d = 63

-1+ 5n - 5 = 63

5n = 69



-1 + (n - 1)5 = 63

n = = 1369 5

45

Number of terms cannot be a fraction.

Thus 63 is not a term of the AP -1, 4, 9, 14, …..



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Arithmetic progressions - Poblem based Arithmetic progressions