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The sum of first four terms of an A. P. is 56 and sum of last four terms is 112. If the first term is 11, then the number of terms is
- a)12
- b)10
- c)1
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
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Given Information
- The sum of the first four terms of an Arithmetic Progression (A.P.) is 56.
- The sum of the last four terms is 112.
- The first term (a) is 11.
Formulas Used
- The sum of the first n terms of an A.P. is given by:
\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\]
where \(d\) is the common difference, and \(n\) is the number of terms.
Calculating the First Four Terms
- For the first four terms:
\[
S_4 = \frac{4}{2} \times (2 \cdot 11 + (4-1)d) = 2 \times (22 + 3d) = 44 + 6d
\]
Setting this equal to 56:
\[
44 + 6d = 56 \implies 6d = 12 \implies d = 2
\]
Calculating the Last Four Terms
- The last four terms can be expressed as:
\[
S_n = a + (n-1)d + a + (n-2)d + a + (n-3)d + a + (n-4)d = 4a + (n-1 + n-2 + n-3 + n-4)d
\]
This simplifies to:
\[
4a + (4n - 10)d = 4 \cdot 11 + (4n - 10) \cdot 2
\]
Setting this equal to 112:
\[
44 + 8n - 20 = 112 \implies 8n + 24 = 112 \implies 8n = 88 \implies n = 11
\]
Conclusion
The calculations show that the number of terms \(n\) is 11. However, since we are not finding any option corresponding to 11, the correct answer is indeed **option D: none of these**.
- The sum of the first four terms of an Arithmetic Progression (A.P.) is 56.
- The sum of the last four terms is 112.
- The first term (a) is 11.
Formulas Used
- The sum of the first n terms of an A.P. is given by:
\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\]
where \(d\) is the common difference, and \(n\) is the number of terms.
Calculating the First Four Terms
- For the first four terms:
\[
S_4 = \frac{4}{2} \times (2 \cdot 11 + (4-1)d) = 2 \times (22 + 3d) = 44 + 6d
\]
Setting this equal to 56:
\[
44 + 6d = 56 \implies 6d = 12 \implies d = 2
\]
Calculating the Last Four Terms
- The last four terms can be expressed as:
\[
S_n = a + (n-1)d + a + (n-2)d + a + (n-3)d + a + (n-4)d = 4a + (n-1 + n-2 + n-3 + n-4)d
\]
This simplifies to:
\[
4a + (4n - 10)d = 4 \cdot 11 + (4n - 10) \cdot 2
\]
Setting this equal to 112:
\[
44 + 8n - 20 = 112 \implies 8n + 24 = 112 \implies 8n = 88 \implies n = 11
\]
Conclusion
The calculations show that the number of terms \(n\) is 11. However, since we are not finding any option corresponding to 11, the correct answer is indeed **option D: none of these**.
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The sum of first four terms of an A. P. is 56 and sum of last four terms is 112. If the first term is 11, then the number of terms isa)12b)10c)1d)none of theseCorrect answer is option 'D'. Can you explain this answer?
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The sum of first four terms of an A. P. is 56 and sum of last four terms is 112. If the first term is 11, then the number of terms isa)12b)10c)1d)none of theseCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The sum of first four terms of an A. P. is 56 and sum of last four terms is 112. If the first term is 11, then the number of terms isa)12b)10c)1d)none of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of first four terms of an A. P. is 56 and sum of last four terms is 112. If the first term is 11, then the number of terms isa)12b)10c)1d)none of theseCorrect answer is option 'D'. Can you explain this answer?.
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