Question

1+2+3+---------+n/1+3+5+-------+(2n-1) what is the value​

Answer 1

Answer:

$\frac{(n+1)}{2n}$

Step-by-step explanation:

[ 1+2+3+---------+n ] / [ 1+3+5+-------+(2n-1) ]

= $\frac{n(n+1)}{2}$ / n²

= $\frac{n(n+1)}{2n^2}$

= $\frac{(n+1)}{2n}$

Answer 2

Let P(n): 1 + 3 + 5 +  + (2n - 1) = n  

2

 be the given statement

Step 1: Put n = 1

Then, L.H.S = 1

R.H.S = (1)  

2

 = 1

∴. L.H.S = R.H.S.

⇒ P(n) istrue for n = 1

Step 2: Assume that P(n) istrue for n = k.

∴ 1 + 3 + 5 ++ (2k - 1) = k  

2

 

Adding 2k + 1 on both sides, we get

1 + 3 + 5+ (2k - 1) + (2k + 1) = k  

2

 + (2k + 1) = (k + 1)  

2

 

∴ 1 + 3 + 5 ++ (2k -1) + (2(k + 1) - 1) = (k + 1)  

2

 

⇒ P(n) is true for n = k + 1.

∴ by the principle of mathematical induction P(n) is true for all natural numbers 'n'

Hence, 1 + 3 + 5 ++ (2n - 1) =n  

2

, for all n ϵ n