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Consider the recurrence relation a1=4, an=5n+an-1. The value of a64 is _________
- a)10399
- b)23760
- c)75100
- d)53700
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Consider the recurrence relation a1=4, an=5n+an-1. The value of a64is ...
Given: a1=4, an=5n an-1
To find: the value of a64
Method:
We can find the value of a64 by using the given recurrence relation recursively.
- Using the recurrence relation, we can find the value of a2 as follows:
a2 = 5(2) a1 = 5(2)(4) = 40
- Similarly, we can find the value of a3 as follows:
a3 = 5(3) a2 = 5(3)(40) = 600
- Continuing in this manner, we can find the value of an for any n.
- Now, to find the value of a64, we can use the recurrence relation repeatedly until we reach the desired value:
a64 = 5(64) a63 = 5(64)(5(63) a62) = 5(64)(5(63) (5(62) a61)) = … = 5(64)(5(63) … (5(2) a1))
- Substituting the value of a1 = 4, we get:
a64 = 5(64)(5(63) … (5(2) 4)) = 5(64)(5(63) … (5(2)))4
- We can simplify the expression by recognizing that the product 5(63) … (5(2)) is simply 5 raised to the power of the sum of the integers from 2 to 63, which can be calculated using the formula for the sum of an arithmetic series:
5 + 10 + 15 + … + 310 = (63/2)(5 + 310) = 1980
- Substituting this value, we get:
a64 = 5(64)(51980)4 = 103994
Therefore, the value of a64 is 10399.
Answer: (A) 10399
To find: the value of a64
Method:
We can find the value of a64 by using the given recurrence relation recursively.
- Using the recurrence relation, we can find the value of a2 as follows:
a2 = 5(2) a1 = 5(2)(4) = 40
- Similarly, we can find the value of a3 as follows:
a3 = 5(3) a2 = 5(3)(40) = 600
- Continuing in this manner, we can find the value of an for any n.
- Now, to find the value of a64, we can use the recurrence relation repeatedly until we reach the desired value:
a64 = 5(64) a63 = 5(64)(5(63) a62) = 5(64)(5(63) (5(62) a61)) = … = 5(64)(5(63) … (5(2) a1))
- Substituting the value of a1 = 4, we get:
a64 = 5(64)(5(63) … (5(2) 4)) = 5(64)(5(63) … (5(2)))4
- We can simplify the expression by recognizing that the product 5(63) … (5(2)) is simply 5 raised to the power of the sum of the integers from 2 to 63, which can be calculated using the formula for the sum of an arithmetic series:
5 + 10 + 15 + … + 310 = (63/2)(5 + 310) = 1980
- Substituting this value, we get:
a64 = 5(64)(51980)4 = 103994
Therefore, the value of a64 is 10399.
Answer: (A) 10399
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Community Answer
Consider the recurrence relation a1=4, an=5n+an-1. The value of a64is ...
an=5n+an-1
= 5n + 5(n-1) + … + an-2
= 5n + 5(n-1) + 5(n − 2) +…+ a1
= 5n + 5(n-1) + 5(n − 2) +…+ 4 [since, a1=4]
= 5n + 5(n-1) + 5(n − 2) +…+ 5.1 – 1
= 5(n + (n − 1)+…+2 + 1) – 1
= 5 * n(n+1)/ 2 – 1
an = 5 * n(n+1)/ 2 – 1
Now, n=64 so the answer is a64 = 10399.
= 5n + 5(n-1) + … + an-2
= 5n + 5(n-1) + 5(n − 2) +…+ a1
= 5n + 5(n-1) + 5(n − 2) +…+ 4 [since, a1=4]
= 5n + 5(n-1) + 5(n − 2) +…+ 5.1 – 1
= 5(n + (n − 1)+…+2 + 1) – 1
= 5 * n(n+1)/ 2 – 1
an = 5 * n(n+1)/ 2 – 1
Now, n=64 so the answer is a64 = 10399.
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