Question

1/2.5 +1/5.8 +1/8.11 +..... +1/(3n-1) (3n+2)= n/(6n+4) prove the following by using the principle of mathematical induction for all n belongs N

Answer 1

Answer:

Step-by-step explanation:

Given  

1/2.5 +1/5.8 +1/8.11 +..... +1/(3n-1) (3n+2)= n/(6n+4) prove the following by using the principle of mathematical induction for all n belongs N  

Let Sn = 1 / 2.5 + 1 / 5.8 + 1 / 8.11 +……..+ 1 / (3n – 1)(3n + 2) = n / (6n + 4)

Now to prove for n = 1

1 / 2 x 5 = 1/10

1/6 + 4 = 1/10

So it is true for n = 1

Now assume sn is true for n = k

1/2.5 + 1/5.8 + 1/8.11 ++………..+1/(3k – 1)(3k + 2) = k / (6k + 4)

Now Sn for n = k + 1

1/2.5 + 1/5.8 + 1/8.11 + …………+ 1/3(k + 1) – 1)(3(k + 1) + 2)

k/(6k + 4) + 1/(3k + 2)(3k + 5)

k/2(3k + 2) + 1/(3k + 2)(3k + 5)

k(3k + 5) + 2 / 2(3k + 2)(3k + 5)

3k^2 + 5k + 2 / 2 (3k + 2)(3k + 5)

(3k + 2)(k + 1) / 2(3k + 2)(3k + 5)

= (k + 1) / (6k + 10)

= (k + 1) / 6 (k + 1) + 4 = R. H. S

So it is true for n = k + 1

by using the principle of mathematical induction it is true for all natural numbers.