Find sum of the series and simplify the polynomial?

Question

i- find sum of the series 4,7,10,.....28 and 1/2,1/8,1/32,1/128,1/152 
ii- simplify the polynopmial 
a- x^2+3x+5/(x-2)(x+3)(x-5) 
b- 4x^3+5x^2-4x+3/(x+2) 
c- 1/(x-3)(x+2)^3

Answer 1

i) Series is 4,7,10,13,16,19,22,25,28        [ Since differene is 3 between the two terms ]

  Sum of the series is 4+7+10+13+16+19+22+25+28 = 144.

  The sum of the series 4,7,10,13,16,19,22,25,28 is 144.

   Series is 1/2, 1/8, 1/32, 1/28, 1/52

   Sum of the series is 1/2) + (1/8) + (1/32) + (1/128) + (1/152)

                                 = 1631/2432

                                 =~ 0.67064

   The sum of the series 1/2, 1/8, 1/32, 1/28, 1/52 is 0.67064.
                                                               

Comments

Sequence : An ordered list of numbers (or) A sequence is a list of numbers in a particular order.

Series : The sum of an ordered list of numbers (or) A series is an indicated sum of the terms of a sequence.

Arithmetic Sequence : A sequence of numbers with a common difference between any two consecutive terms.

Arithmetic Series : The sum of terms in an arithmetic sequence.

Geometric Sequence : A sequence of numbers with a common ratio or multiplier between any two consecutive terms.

Arithmetic Series : The sum of terms in an geometric sequence.

The numbers are 4, 7, 10, . . . . , 28.

The above numbers are follow the particular order, so this is a sequence.

The above sequence is arithmetic sequence, since common difference is 3 (7 - 4 = 3, 10 - 7 = 3).

The sum S_n of the first n terms of an arithmetic series is given by , where t₁ = first term = 2, t_n = last term = 28 and d = common difference = 3.

Find the value of n.

Substitute the value of t₁ = 4, t_n = 28 and d = 3 in nth term of arithmetic sequence : t_n = t₁ + (n - 1)d.

28 = 4 + (n - 1)3

24 = 3n - 3

27 = 3n

9 = n.

Find the value of s₉.

Substitute the value of n = 9, t₁ = 4 and t_n = t₉ = 28 in the sum S_n of the first n terms of an arithmetic series : S_n = (n/2) [t₁ + t_n].

S₉ = (9/2) [4 + 28]

S₉ = (9/2) [32]

S₉ = (9) [16]

S₉ = 144.

Let assume that the numbers are 1/2, 1/8, 1/32, 1/128, 1/512.

The above numbers are follow the particular order, so this is a sequence.

The above sequence is geometric sequence, since common ratio is 1/4 [ (1/8)/(1/2) = (1/32)/(1/8) = (1/128/(1/32) = (1/152)/(1/512) = 1/4 ].

The sum S_n of the first n terms of a geometric series is given by , where r  ≠ 1, t₁ = first term = 1/2, r = common ratio = 1/4.

Find the value of n.

Substitute the value of t_n = 1/512, t₁ = 1/2 and r = 1/4 in nth term of geometric sequence : t_n = t₁ · r^{n - 1}.

1/512 = 1/2 · (1/4)^{n - 1}.

1/256 = (1/4)^{n - 1}

(1/4)⁴ = (1/4)^{n - 1}

4 = n - 1

5 = n.

Find the value of s₅.

Substitute the value of n = 5, t₁ = 1/2 and r = 1/4 in the sum S_n of the first n terms of a geometric series : .

.

Answer 2

ii) a)

 

is the answer.

 

b)

 

is the answer.

 

c) 1/(x-3)(x+2)^3

The above expression is already in the simplified form. So, there is no chance to simplify it further.