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t is an number in the series: 1/t + 3/t + 5/t + ...+ t-1/t a)Determine the number of terms in the series in terms of t. b)Determine the sum of the series in terms of t. c)Hence,or otherwise,evaluate:(1/4+3/4)+(1/6+3/6+5/6)+(1/8+3/8+5/8+7/8)+...+(1/50+3/50+5/50+49/50)​

Respuesta :

Answer:

Below in bold.

Step-by-step explanation:

1/t + 3/t + 5/t +  ...+  t-1/t

This is an arithmetic series in which the common difference is 2/t.

a) The number of terms = [ (last term - first term) / common difference] + 1

=  [(t-1)/t - 1/t) / 2/t]  + 1

= (t - 2)/t * t/2 + 1

= (t -2)/2 + 1

= (t-2) / 2 + 2/2

= t/2.

b). Sum of the series (n/2)(a + l)  where a = first term , n = number of terms and l = last term, so

Sum = (t/2)/2 ( 1/t + (t-1)/t)

= t/4 * 1

= t/4.

c). (1/4+3/4 )+ (1/6+3/6+5/6 )+( 1/8+3/8+5/8+7/8) +...+(1/50+3/50+5/50+49/50)​

Using the results from a) and b):

The sum = 4/4  + 6/4 + 8/4 + ......+ 50/4.

= 1 + 1.5 + 2 +...... + 12.5

This is an AS  with common difference = 0.5,

Number of terms = [(12.5 -1) / 0.5] + 1

= 23 + 1

= 24, so

Sum = (24/2)(1 + 12.5)

=  12  * 13.5

=  162.

raphealnwobi

The number of terms is t/2, and the sum of the series is 1/2 - (t - 2)/4

What is an arithmetic sequence?

In an arithmetic sequence, the common difference is the difference between successive terms.

a) From the sequence:

a = 1/t; d = (3/t) - (1/t) = 2/t

The number of terms is:

(t - 1)/t = 1/t + (n - 1)(2/t)

t - 1 = 1 + 2(n - 1)

t = 2n

n = t/2

b) The sum of the series is:

S = (n/2)[2a + (n - 1)d]

S = (t/2/2)[2(1/t) + (t/2 - 1)(2/t)]

S = (t/4)[2/t - (t - 2)/t]

S = 1/2 - (t - 2)/4

The number of terms is t/2, and the sum of the series is 1/2 - (t - 2)/4

Find out more on arithmetic sequence at: https://brainly.com/question/6561461

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