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Consider the recurrence relation a1 = 8, an = 6n2 + 2n + an-1. Let a99 = k x 104. The value of K is _____ Note : This question was asked as Numerical Answer Type.
- a)190
- b)296
- c)198
- d)200
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Consider the recurrence relation a1= 8, an= 6n2+ 2n + an-1. Let a99= k...
To solve the given recurrence relation and find the value of k in a99 = k x 104, we can use the method of iteration.
1. Finding the pattern:
Let's find the values of a2, a3, a4, and a5 using the given recurrence relation:
a2 = 6(2^2) - 2(2) + a1 = 24 - 4 + 8 = 28
a3 = 6(3^2) - 2(3) + a2 = 54 - 6 + 28 = 76
a4 = 6(4^2) - 2(4) + a3 = 96 - 8 + 76 = 164
a5 = 6(5^2) - 2(5) + a4 = 150 - 10 + 164 = 304
By observing the pattern, we can see that the values of a2, a3, a4, a5 are 28, 76, 164, and 304 respectively.
2. Writing the general term:
From the pattern, we can deduce that the general term of the given recurrence relation can be written as:
an = 6n^2 - 2n + (n-1)^2 + 4
3. Finding the value of a99:
Using the general term, we can find the value of a99:
a99 = 6(99^2) - 2(99) + (99-1)^2 + 4
= 6(9801) - 198 + 98^2 + 4
= 58806 - 198 + 9604 + 4
= 68816
4. Finding the value of k:
Given that a99 = k x 104, we can equate the two expressions:
68816 = k x 104
k = 68816 / 104
k ≈ 661.538
Rounded to the nearest whole number, k ≈ 662.
Therefore, the value of k is approximately 662, which corresponds to option C.
1. Finding the pattern:
Let's find the values of a2, a3, a4, and a5 using the given recurrence relation:
a2 = 6(2^2) - 2(2) + a1 = 24 - 4 + 8 = 28
a3 = 6(3^2) - 2(3) + a2 = 54 - 6 + 28 = 76
a4 = 6(4^2) - 2(4) + a3 = 96 - 8 + 76 = 164
a5 = 6(5^2) - 2(5) + a4 = 150 - 10 + 164 = 304
By observing the pattern, we can see that the values of a2, a3, a4, a5 are 28, 76, 164, and 304 respectively.
2. Writing the general term:
From the pattern, we can deduce that the general term of the given recurrence relation can be written as:
an = 6n^2 - 2n + (n-1)^2 + 4
3. Finding the value of a99:
Using the general term, we can find the value of a99:
a99 = 6(99^2) - 2(99) + (99-1)^2 + 4
= 6(9801) - 198 + 98^2 + 4
= 58806 - 198 + 9604 + 4
= 68816
4. Finding the value of k:
Given that a99 = k x 104, we can equate the two expressions:
68816 = k x 104
k = 68816 / 104
k ≈ 661.538
Rounded to the nearest whole number, k ≈ 662.
Therefore, the value of k is approximately 662, which corresponds to option C.
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Consider the recurrence relation a1= 8, an= 6n2+ 2n + an-1. Let a99= k...
a1 = 8 an = 6n2 + 2n + an-1 an = 6[n2 + (n-1)2] + 2[n + (n-1)] + an-2 Continuing the same way till n=2, we get an = 6[n2 + (n-1)2 + (n-2)2 + ... + (2)2] + 2[n + (n-1) + (n-2) + ... + (2)] + a1 an = 6[n2 + (n-1)2 + (n-2)2 + ... + (2)2] + 2[n + (n-1) + (n-2) + ... + (2)] + 8 an = 6[n2 + (n-1)2 + (n-2)2 + ... + (2)2] + 2[n + (n-1) + (n-2) + ... + (2)] + 6 + 2 an = 6[n2 + (n-1)2 + (n-2)2 + ... + (2)2 + 1] + 2[n + (n-1) + (n-2) + ... + (2) + 1] an = (n)*(n+1)*(2n+1) + (n)(n+1) = (n)*(n+1)*(2n+2) an = 2*(n)*(n+1)*(n+1) = 2*(n)*(n+1)2 Now, put n=99. a99 = 2*(99)*(100)2 = 1980000 = K * 104 Therefore, K = 198. Thus, C is the correct choice.
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