Find the sum of 'n' terms in the series
1.4 + 3.7 + 5.10 + ...............
Answers
Answer:
An iron block of mass 150 g at 30°C was heated till it completely melted. Find the quantity of heat required: 1. to bring the block to its melting point 2. the total heat energy required to melt the block to its melting point 3. total heat energy required to completely melt the iron block (melting point of iron = 1085 °C, specific heat capacity = 0.399) g +0; specific latent heat of fusion, L= 180 Jg=?)
Answer:
In each term ,
there are two factors
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First factors in each term are 1 , 3 , 5 , ..........
These are in A.P.
:. nth term = 1 + (n - 1) 2 [ •.• a = 1 ; d = 2 ]
===> 1 + 2n - 2
===> 2n & 1
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Second factors in each term are 4 , 7 , 10 , ......
:. nth term = 4 + (n - 1) 3
===> 4 + 3n - 3
===> 3n + 1
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:. nth term of the given series = (2n - 1) (3n - 1)
===> 6n² - !n - 1
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Sn = 6 € n² - 1 € n - n
=> 6. n (n + 1) (2n + 1) / 6 - n (n + 1) / 2 - n
=> 2n (n + 1) (2n + 1) - n(n + 1) - 2n / 2
=> n [2(2n² + 3n + 1) - n - 1 - 2] / 2
=> n/2 [ 4n² + 6n + 2 - n - 1 - 2 ]
=> n/2 [ 4n² + 5n - 1 ]
Sum of the 'n' terms of the given series = n/2 [4n² + 5n - 1]