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26 Jul 2023, 08:00
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D
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Difficulty:
95%
(hard)
Question Stats:
44% (02:25) correct
56%
(02:37)
wrong
based on 561
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The infinite sequence A is such that \(a_1=3\) and \(a_{n} = a_{n-1} + 3*10^{n-1}\), for all n > 1. If each term in the product \(x_1*x_2*x_3*...*x_k\) is taken from the infinite sequence A, repetitions allowed, and the product equals 173,446,418,443,770,747, which of the following could be equal to k?
A. 16
B. 17
C. 18
D. 19
E. 20
A. 16
B. 17
C. 18
D. 19
E. 20
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26 Jul 2023, 08:37
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given that
a1= 3
a2 will be 33
a3 = 333
a4=3333
product of values is 173,446,418,443,770,747,
last digit is 7
cyclicity of 3 is (4)
3^1 =3
3^2=9
3^3=7
3^4=1
value of k must be 19 to get the product value ending with digit 7
option D is correct
a1= 3
a2 will be 33
a3 = 333
a4=3333
product of values is 173,446,418,443,770,747,
last digit is 7
cyclicity of 3 is (4)
3^1 =3
3^2=9
3^3=7
3^4=1
value of k must be 19 to get the product value ending with digit 7
option D is correct
Bunuel
General Discussion
26 Jul 2023, 08:42
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Let's calculate the first few terms of the sequence A to get a better understanding:
a1 = 3
a2 = a1 + 3*10^1 = 3 + 30 = 33
a3 = a2 + 3*10^2 = 33 + 300 = 333
a4 = a3 + 3*10^3 = 333 + 3000 = 3333
and so on...
We can see that each term in the sequence A is a string of 3's, the length of which is equal to the term's position in the sequence.
Now, we are given that the product of some terms from the sequence A equals 173,446,418,443,770,747. We are asked to find the number of terms in this product, which is represented by k.
Since unit digit is 7 and we know unit digit in power series of 3 is : 3,9,7,1
To have unit digit as 7, we need k of form 4p+3
So 19 is the correct answer
Answer is D
a1 = 3
a2 = a1 + 3*10^1 = 3 + 30 = 33
a3 = a2 + 3*10^2 = 33 + 300 = 333
a4 = a3 + 3*10^3 = 333 + 3000 = 3333
and so on...
We can see that each term in the sequence A is a string of 3's, the length of which is equal to the term's position in the sequence.
Now, we are given that the product of some terms from the sequence A equals 173,446,418,443,770,747. We are asked to find the number of terms in this product, which is represented by k.
Since unit digit is 7 and we know unit digit in power series of 3 is : 3,9,7,1
To have unit digit as 7, we need k of form 4p+3
So 19 is the correct answer
Answer is D
