Rd Chapter 9 Arithmetic Progressions Ex 9.4 Solutions
Question - 11 : - The 26th, 11th and last term of an A.P. are 0, 3 and – 1/5 , respectively. Find the common difference and the number of terms. [NCERT Exemplar]
Answer - 11 : -
Let the first term, common difference and number of terms of an A.P. are a, d and n, respectively.
We know that, if last term of an A.P. is known, then
l = a + (n – 1) d ……(i)
and nth term of an A.P is
Question - 12 : - If the nth term of the A.P. 9, 7, 5, … is same as the nth term of the A.P. 15, 12, 9, … find n.
Answer - 12 : - In A.P 9, 7, 5, ………
Here first term (a) = 9 and d = 7 – 9 = -2 {or 5 – 7 = -2}
nth term (an) = a + (n – 1) d = 9 + (n – 1) (-2) = 9 – 2n + 2 = 11 –2n
Now in A.P. 15, 12, 9, …..
Here first term (a) = 15 and (d) = 12 – 15 = -3
nth term (an) = a + (n – 1) d = 15 + (n – 1) x (-3)
The nth term of first A.P. = nth term of second A.P.
11 – 2n = 18 – 3n
=> -2n + 3n = 18 – 11
=> n = 7
Hence n = 7
Question - 13 : - Find the 12th term from the end of the following arithmetic progressions :
(i) 3, 5, 7, 9, … 201
(ii) 3, 8, 13,…, 253
(iii) 1, 4, 7, 10, …, 88
Answer - 13 : -
(i) In the A.P. 3, 5, 7, 9, … 201
First term (a) = 3, last term (l) = 201
and common difference (d) = 5 – 3 = 2
We know that nth term from the last = l – (n – 1 ) d
12th term from the last = 201 – (12 – 1) x 2 = 201 – 11 x 2 = 201 – 22 = 179
(ii) In the A.P. 3, 8, 13, …, 253
First term (a) = 3
Common difference (d) = 8 – 3 = 5
and last term = 253
The nth term from the last = l – (n – 1) d
12th term from the last = 253 – (12 – 1) x 5 = 253 – 11 x 5 = 253 – 55 = 198
(iii) In the A.P. 1, 4, 7, 10, …, 88
First term (a) = 1
Common difference (d) = 4 – 1 = 3
and last term = 88
The nth term from the last = l – (n – 1) d
12th term from the last = 88 – (12 – 1) x 3 = 88 – 11 x 3 = 88 – 33 = 55
Question - 14 : - The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
Answer - 14 : -
Question - 15 : - Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
Answer - 15 : - In an A.P.
6th term (a6) = 12
and 8th term (a8) = 22
Let a be the first term and d be the common difference, then
Question - 16 : - How many numbers of two digit are divisible by 3 ?
Answer - 16 : -
Let n be the number of terms which are divisible by 3 and d are of two digit numbers
Let a be the first term and d be the common difference, then
a = 12, d = 3, last term = 99
an = a + (n – 1) d
99 = 12 + (n – 1) x 3
=> 99 = 12 + 3n – 3
=> 3n = 99 – 9
=> n = 30
Number of terms = 30
Question - 17 : - An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
Answer - 17 : -
In an A.P.
n = 60
First term (a) = 7 and last term (l) = 125
Let d be the common difference, then
a60 = a + (60 – 1) d
=> 125 = 7 + 59d
=> 59d = 125 – 7 = 118
Common difference = 2
Now 32nd term (a32) = a + (32 – 1) d = 7 + 31 x 2 = 7+ 62 = 69
Question - 18 : - The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
Answer - 18 : -
Question - 19 : - The first term of an A.P. is 5 and its 100th term is -292. Find the 50th term of this A.P.
Answer - 19 : -
First term of an A.P. = 5
and 100th term = -292
Question - 20 : - Find a30 – a20 for the A.P.
(i) -9, -14, -19, -24, …
(ii) a, a + d, a + 2d, a + 3d, …
Answer - 20 : -