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# Let Sn Denote the Sum of N Terms of an A.P. Whose First Term is A. If the Common Difference D is Given by D = Sn − Ksn−1 + Sn−2, Then K = - Mathematics

MCQ

Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by Sn − kSn−1 + Sn−2, then k =

#### Options

• 1

• 2

• 3

• none of these.

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#### Solution

In the given problem, we are given  d = Sn - kSn-1 + Sn-2

We need to find the value of k

So here,

First term = a

Common difference = d

Sum of terms = Sn

Now, as we know,

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

Also, for n-1 terms,

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

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

Further, for n-2 terms,

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

= (n-2)/2 [ 2a + ( n - 3) d ]           ...............(3)

Now, we are given,

d = Sn  - kSn -1 + Sn-2

d  +  KSn-1 = Sn + Sn-2

 K = (S_n + S_(n-2) - d ) /S_(n-1)

Using (1), (2) and (3) in the given equation, we get

k = (n/2[2a + (n-1) d ] + (n-2)/2 [2a + ( n - 3)d]-d)/((n-1)/2 [ 2a + (n - 2 ) d ])

Taking 1/2 common, we get,

K = (n[2a + (n-1)d] + (n-2) [2a + ( n- 3)d]- 2d)/((n-1)[2a + (n-2) d])

 =(2an + n^2d - nd + 2an + n^2d - 3nd - 4a - 2nd + 6d - 2d)/(2an + n^2d - 2nd - 2a - nd + 2d)

 = (2n^2d + 4an - 6nd - 4a + 4d)/(n^2d + 2an - 3nd + -2a + 2d)

Taking 2 common from the numerator, we get,

k = (2(n^2 d + 2an - 3nd  + -2a + 2d))/(n^2d + 2an - 3nd + -2a +2d)

= 2

Therefore, k = 2

Is there an error in this question or solution?
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Q 9 | Page 57
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