Which of the following sequences are in G.P.? (ii) 4,44,444,4444,...

Which of the following sequences are in G.P.?

(ii) 4,44,444,4444,...

Solution :

First term (a)  = 3, common ratio

r  = t2/t1   =  44/4 r = 11

r  = t3/t2   =  444/44 r = 111/11

r  = t2/t1   =  4444/444 r = 1111/111

Since the common ratios are not same, it is not geometric sequence.

If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm = (mn/2)(1 + mn)

If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm  = (mn/2)(1 + mn)

Solution :

n = [(l-a)/d] + 1

 n  = [((2mn - n + 1) - (1 + n))/2n] + 1

 n  = [(2mn - n + 1 - 1 - n)/2n] + 1

n = [(2mn - 2n)/2n] + 1

n = (m - 1) + 1

n = m

 S1 + S2 + S3 +............Sm  

Hence proved.

Find the 19th term of an A.P. -11,-15,-19,...

Find the 19th term of an A.P. -11,-15,-19,...

Solution :

nth term of the A.P

tn  = a + (n - 1)d

a = -11 , d = -15 - (-11)  = -15 + 11 ==> -4

t19  = -11 + (19 - 1)(-4)

=  -11 + 18(-4)

=  - 11 - 72

t19  =   -83

Therefore, the 19th term of the A.P. is -83. 

Check whether the following sequences are in A.P. (iii) 9, 13, 17, 21, 25,...

Check whether the following sequences are in A.P.

(iii) 9, 13, 17, 21, 25,...

Solution :

d = t2 - t1 d = 13 - 9 d  = 4

d = t3 - t2 d = 17 - 13 d  = 4

The given sequence is an arithmetic progression.

Check whether the following sequences are in A.P. (i) a - 3, a - 5, a -7,...

Check whether the following sequences are in A.P.

(i) a - 3, a - 5, a -7,...

Solution :

d = t2 - t1 d = (a - 5) - (a - 3)   =  a - 5 - a + 3 d  = -2

d = t3 - t2 d = (a - 7) - (a - 5)   =  a - 7 - a + 5 d  = -2

Since the common difference area same, the given sequence is an arithmetic progression.

Find the nth term of the following sequences (iii) 3, 8, 13, 18,...

Find the nth term of the following sequences

(iii) 3, 8, 13, 18,...

Solution :

  =  3, 8, 13, 18,...

we may see the common things followed in each term.

  =  (5 - 2), (10 - 2), (15 - 2), (20 - 2),...

nth term of the sequence is 5n - 2.

Find the nth term of the following sequences (i) 2, 5, 10, 17,...

Find the nth term of the following sequences

(i) 2, 5, 10, 17,...

Solution :

(1 + 1), (4 + 1), (9 + 1), (16 + 1),...

(12 + 1), (22 + 1), (32 + 1), (42 + 1),...

nth term of the sequence  :

an  = n2 + 1