Question

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by $\frac{l^{2}-a^{2}} {k-(l+a)'}$ , then k =

Answer 1

Answer:

The value of k is 2S.

Among the given options option (b) 2S is a correct answer.

Step-by-step explanation:

Given :  

The first and last term of an A.P. are 'a' and 'l' .  S is the sum of all the terms of the A.P. and the common difference 'd' = (l² - a²)/k - (l + a)

By using the formula ,an = l = a + (n - 1)d

n = (l - a)/d + 1………(1)

By using the formula ,Sum of nth terms , Sn = n/2 [2a + (n – 1) d]

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

[from eq 1]

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

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

2S/[a + l ] =  [(l - a)/d + 1]

[2S/(a + l )] - 1 =  [(l - a)/d]

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

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

d = (l² - a²)/ [2S - (a + l)]  ................(2)

On Comparing the value of d in eq 2 with the given value of 'd' = (l² - a²)/k - (l + a) , we get

k = 2S

Hence, the value of k is 2S.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

Answer is in the attachment