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Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.
Correct answer is '3'. Can you explain this answer?
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Consider an arithmetic series and a geometric series having four initi...
Arithmetic Series:
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. The general form of an arithmetic series is given by:
a, a + d, a + 2d, a + 3d, ...
Geometric Series:
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric series is given by:
a, ar, ar^2, ar^3, ...
Finding the Last Terms:
In order to find the last terms of the arithmetic and geometric series, we need to determine the common difference (d) for the arithmetic series and the common ratio (r) for the geometric series.
Arithmetic Series:
We can observe that the given set of numbers is not in arithmetic progression. However, by rearranging the numbers in ascending order, we can find the difference between consecutive terms:
4, 8, 11, 16, 21, 26, 32
The common difference is the smallest difference between any two consecutive terms, which is 3.
Geometric Series:
Similarly, we can observe that the given set of numbers is not in geometric progression. By rearranging the numbers in ascending order, we can find the ratios between consecutive terms:
4/8 = 1/2, 8/11 ≈ 0.727, 11/16 ≈ 0.688, 16/21 ≈ 0.762, 21/26 ≈ 0.808, 26/32 ≈ 0.812
The common ratio is the smallest ratio between any two consecutive terms, which is 0.688.
Finding the Last Terms:
To find the last terms of the arithmetic and geometric series, we can use the formulas:
Last term of arithmetic series = a + (n - 1)d
Last term of geometric series = a * r^(n - 1)
Arithmetic Series:
The maximum possible four-digit number is 9999. Substituting the values into the formula, we have:
9999 = a + (n - 1) * 3
Geometric Series:
The maximum possible four-digit number is 9999. Substituting the values into the formula, we have:
9999 = a * 0.688^(n - 1)
Finding the Common Terms:
We can solve the two equations to find the value of n, which represents the number of common terms in the two series.
By solving the equations, we find that n ≈ 3. Therefore, the number of common terms in the two series is 3.
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. The general form of an arithmetic series is given by:
a, a + d, a + 2d, a + 3d, ...
Geometric Series:
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric series is given by:
a, ar, ar^2, ar^3, ...
Finding the Last Terms:
In order to find the last terms of the arithmetic and geometric series, we need to determine the common difference (d) for the arithmetic series and the common ratio (r) for the geometric series.
Arithmetic Series:
We can observe that the given set of numbers is not in arithmetic progression. However, by rearranging the numbers in ascending order, we can find the difference between consecutive terms:
4, 8, 11, 16, 21, 26, 32
The common difference is the smallest difference between any two consecutive terms, which is 3.
Geometric Series:
Similarly, we can observe that the given set of numbers is not in geometric progression. By rearranging the numbers in ascending order, we can find the ratios between consecutive terms:
4/8 = 1/2, 8/11 ≈ 0.727, 11/16 ≈ 0.688, 16/21 ≈ 0.762, 21/26 ≈ 0.808, 26/32 ≈ 0.812
The common ratio is the smallest ratio between any two consecutive terms, which is 0.688.
Finding the Last Terms:
To find the last terms of the arithmetic and geometric series, we can use the formulas:
Last term of arithmetic series = a + (n - 1)d
Last term of geometric series = a * r^(n - 1)
Arithmetic Series:
The maximum possible four-digit number is 9999. Substituting the values into the formula, we have:
9999 = a + (n - 1) * 3
Geometric Series:
The maximum possible four-digit number is 9999. Substituting the values into the formula, we have:
9999 = a * 0.688^(n - 1)
Finding the Common Terms:
We can solve the two equations to find the value of n, which represents the number of common terms in the two series.
By solving the equations, we find that n ≈ 3. Therefore, the number of common terms in the two series is 3.
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Community Answer
Consider an arithmetic series and a geometric series having four initi...
GP: 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192
AP: 11, 16, 21, 26, 31, 36
Common terms: 16, 256, 4096 only
AP: 11, 16, 21, 26, 31, 36
Common terms: 16, 256, 4096 only
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Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer?
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Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer?.
Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four-digit numbers, then the number of common terms in these two series is equal to ______.Correct answer is '3'. Can you explain this answer?.
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