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The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?
- a)6
- b)9
- c)12
- d)16
- e)18
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for a...
In complex sequence questions, the best strategy usually is to look for a pattern in the sequence of terms that will allow you to avoid having to compute every term in the sequence.
In this case, we know that the first term of Sk is 1 and the first term of An is 9. So when n = 1 and k = 1, q = 9 + 1 = 10 and the sum of the digits of q is 1 + 0 = 1.
since S1= 1, S2= (10)(1) +(2) = 10+2 = 12. since A1 =9,
A2= (10) (9)+(9-(2-1)) = 90+(9-1) = 90+8 = 98. so When K=2 and n = 2, q = 12 + 98 = 110 and the sum of the digits of q is 1 + 1 + 0 = 2.
Since S2= 12, S3 =(10)(12)+3= 120+3= 123. Since A2= 98, It is true that
A3= (10)(98) + (9-(3-1)) = 980+7 = 987. So when n = 3 and k = 3, q = 123 + 987 = 1110 and the sum of the digits of q is 1 + 1 + 1 + 0 = 3.
At this point, we can see a pattern: sk proceeds as 1, 12, 123, 1234..., and An proceeds as 9, 98, 987, 9876.... The sum q therefore proceeds as 10, 110, 1110, 11110... The sum of the digits of q, therefore, will equal 9 when q consists of nine ones and one zero. Since the number of ones in q is equal to the value of n and k (when n and k are equal to each other), the sum of the digits of q will equal 9 when n = 9 and k = 9:
S9 = 123456789 and A9 = 987654321. By way of illustration:
since S1= 1, S2= (10)(1) +(2) = 10+2 = 12. since A1 =9,
A2= (10) (9)+(9-(2-1)) = 90+(9-1) = 90+8 = 98. so When K=2 and n = 2, q = 12 + 98 = 110 and the sum of the digits of q is 1 + 1 + 0 = 2.
Since S2= 12, S3 =(10)(12)+3= 120+3= 123. Since A2= 98, It is true that
A3= (10)(98) + (9-(3-1)) = 980+7 = 987. So when n = 3 and k = 3, q = 123 + 987 = 1110 and the sum of the digits of q is 1 + 1 + 1 + 0 = 3.
At this point, we can see a pattern: sk proceeds as 1, 12, 123, 1234..., and An proceeds as 9, 98, 987, 9876.... The sum q therefore proceeds as 10, 110, 1110, 11110... The sum of the digits of q, therefore, will equal 9 when q consists of nine ones and one zero. Since the number of ones in q is equal to the value of n and k (when n and k are equal to each other), the sum of the digits of q will equal 9 when n = 9 and k = 9:
S9 = 123456789 and A9 = 987654321. By way of illustration:
When n > 9 and k > 9, the sum of the digits of q is not equal to 9 because the pattern of 10, 110, 1110..., does not hold past this point and the additional digits in q will cause the sum of the digits of q to exceed 9.
Therefore, the maximum value of k + n (such that the sum of the digits of q is equal to 9) is 9 + 9 = 18.
The correct answer is E.
Therefore, the maximum value of k + n (such that the sum of the digits of q is equal to 9) is 9 + 9 = 18.
The correct answer is E.
Most Upvoted Answer
The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for a...
Understanding the given sequences:
- The sequence Sk is given by Sk = 10Sk - 1 + k for k > 1.
- The sequence An is given by An = 10An - 1 + (A1 - (n - 1)) for n > 1.
Given initial values:
- S1 = 1
- A1 = 9
Finding the sum q:
- The sum q is defined as the sum of Sk and An.
- q = Sk + An
Finding the sum of the digits of q:
- To find the sum of the digits of q equal to 9, we need to analyze the sequences Sk and An and find the values of k and n that satisfy this condition.
Finding the maximum value of k + n:
- To maximize k + n, we need to find the values of k and n that satisfy the condition of sum of digits of q being 9.
- By analyzing the sequences and applying the given conditions, we can find the maximum value of k + n.
Therefore, after analyzing the sequences and applying the given conditions, we can determine that the maximum value of k + n when the sum of the digits of q is equal to 9 is 18. Hence, the correct answer is option E.
- The sequence Sk is given by Sk = 10Sk - 1 + k for k > 1.
- The sequence An is given by An = 10An - 1 + (A1 - (n - 1)) for n > 1.
Given initial values:
- S1 = 1
- A1 = 9
Finding the sum q:
- The sum q is defined as the sum of Sk and An.
- q = Sk + An
Finding the sum of the digits of q:
- To find the sum of the digits of q equal to 9, we need to analyze the sequences Sk and An and find the values of k and n that satisfy this condition.
Finding the maximum value of k + n:
- To maximize k + n, we need to find the values of k and n that satisfy the condition of sum of digits of q being 9.
- By analyzing the sequences and applying the given conditions, we can find the maximum value of k + n.
Therefore, after analyzing the sequences and applying the given conditions, we can determine that the maximum value of k + n when the sum of the digits of q is equal to 9 is 18. Hence, the correct answer is option E.
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The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer?
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The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer?.
The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer?.
Solutions for The infinite sequence Sk is defined as Sk = 10 Sk – 1 + k, for all k > 1. The infinite sequence An is defined as An = 10 An – 1 + (A1 – (n - 1)), for all n > 1. q is the sum of Sk and An. If S1 = 1 and A1 = 9, and if An is positive, what is the maximum value of k + n when the sum of the digits of q is equal to 9?a)6b)9c)12d)16e)18Correct answer is option 'E'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT.
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