Sum up to n terms and infinity
 

a)

b)

c)

d)

Find (1)the arithmetic series (2) the harmonic series of n terms of which a and b are the first and last terms and hence show that the product of the r^th term of the first series and (n – r + 1)^th term of the second series is ab.

The minimum number of terms from the beginning of the series  so that the sum may exceed 1568 is

a) 25

b) 27

c) 29

d) 28

The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms are alternately positive and negative the first term is

a) 4

b) -4

c) -12

d) 12

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of the progression equals

a)

b)

c)

d)

Let a_1, a₂, a₃ ………. be terms of an arithmetic progression. If =  , p ≠ q then  equals

a)

b)

c)

d)

Let a_1, a₂, a₃ .  .  .  . a_n be in harmonic progression then the expression a₁a₂ + a₂a₃ + .  .  .  .+ a_{n – 1}a_n is equal to

a) (n – 1)( − )

b) n

c) (n −1)

d) n( − )

If   x = ,  y =  , z =  where a, b, c are in arithmetic progression and |a| < 1, |b| < 1, |c| < 1, then x, y and z are in

a) Harmonic Progression

b) Arithemetico–Geometric Progression

c) Arithmetic Progression

d) Harmonic Progression

The sum of the series     + .  .  .  . is

a)

b)

c)

d)

Let T_r be the r^th term of an AP whose first term is a and common difference is d. If for some positive integers m, n, m ≠ n, T_m =  and T_n =  then a – d equals

a) 0

b) 1

c)

d)  

The sum of the first n terms of the series 1² + 2.2²  + 3²  + 2.4² + 5² + 2.6² + … is   when n is even. When n is odd the sum is

a)

b)

c)

d)

The sum of the series  …… is

a)

b)

c)

d)

The sum of the series   …… up to  is equal to

a) 2ln2

b) ln2  1

c) ln2

d) ln

The value of  2^1/4 . 4^1/8 .  8^1/16 . .  .  .  .is

a)

b) 2

c)

d) 4

Fifth term of a geometric progression is 2 then the product of its 9 terms is

a) 256

b) 512

c) 1024

d) None of these

  is equal to

a) lnx

b) x

c)

d) None of these

is equal to

a) ln(x – 1)

b) lnx

c) x

d) None of these

The sum to infinity of the series
1 +   +   +  +  + ………. is

a) 3

b) 4

c) 6

d) 2

If a_n, b_n be two sequences given by
a_n =, b_n =   
for all n ε ℕ. Then  a₁ . a₂ . . a_n is equal to

a) x – y

b)

c)

d)

The value of 2. is

a)

b)

c)

d)

Let S_n =  +  +  . . . . . . . . . +  , n = 1, 2, 3 . . . . . .then S_n is not greater than

a)

b)

c) 2

d) 4

The question has four alternative choices, only one of which is the correct answer. Select one of them which is correct

Statement I     for n ε N, 2ⁿ  > 1 + n
Statement II   GM > HM and (AM) (HM) = (GM)²

a) Statement I is false, Statement II is correct.

b) Statement I is true, Statement II is true. Statement II is a correct explanation for the statement.

c) Statement I is true, Statement II is true. Statement II is not a correct explanation for the statement.

d) Statement I is true, Statement II is false.

The minimum number of terms from the beginning of the series 20 + 22  + 25 + ……. so that the sum may exceed 1568, is

a) 25

b) 27

c) 29

d) 28