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The third term of an A.P is 7 and the seventh term exceeds three times the third term by 2. Find the first term, common difference and the sum of first 20 terms?
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The third term of an A.P is 7 and the seventh term exceeds three times...
Given Information
- The third term of the A.P. is 7.
- The seventh term exceeds three times the third term by 2.
Finding the First Term and Common Difference
- The formula for the nth term of an A.P. is given by:
- Tn = a + (n-1)d
- Where "a" is the first term and "d" is the common difference.
- For the third term (n=3):
- T3 = a + 2d = 7
- For the seventh term (n=7):
- T7 = a + 6d
- Given that T7 = 3 * T3 + 2:
- T7 = 3 * 7 + 2 = 21 + 2 = 23
- Therefore, a + 6d = 23
Setting Up Equations
- We have two equations:
- 1. a + 2d = 7
- 2. a + 6d = 23
Solving the Equations
- Subtract equation 1 from equation 2:
- (a + 6d) - (a + 2d) = 23 - 7
- 4d = 16
- d = 4
- Substitute d back into equation 1:
- a + 2(4) = 7
- a + 8 = 7
- a = -1
Finding the Sum of the First 20 Terms
- The sum formula for the first n terms of an A.P. is:
- Sn = n/2 * (2a + (n-1)d)
- For the first 20 terms (n=20):
- S20 = 20/2 * (2*(-1) + (20-1)*4)
- S20 = 10 * (-2 + 76)
- S20 = 10 * 74 = 740
Final Results
- First term (a): -1
- Common difference (d): 4
- Sum of the first 20 terms (S20): 740
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In the 2006 Common Wealth games, there was participation from five different countries - USA, China, UK, Australia and Japan. From each of these five countries there were athletes participating in one of the five following events: shot put, javelin, high jump, long jump and marathon. Each participant was allowed to participate in exactly one of the five events.Further it is known that there were at least two participants and at most ten participants from each country and the total number of participants from all the five countries put together was a perfect square.The following information is also available:I. The number of participants in javelin was exactly half the number of participants in each of the other events.II. USA sent an equal number of participants in shot put, high jump and long jump.III. The maximum number of participants sent by Japan is for the long jump event and each country sent at least one participant for the long jump and shot put events.IV. The number of participants from China was two more than that from USA, which in turn was one more than that from UK, which in turn was one more than that from Australia which in turn was one more than that from Japan.V. Except Japan, all the other countries sent at least one participant for the javelin event while Japan sent at least one participant in each of the remaining events.VI. The number of participants sent by China for each of the events was different and there were no participants from China for the high jump event.VII. The number of participants in the marathon event sent by all the other countries other than China is equal.Q. How many countries did not send even a single participant in any one event?

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The third term of an A.P is 7 and the seventh term exceeds three times the third term by 2. Find the first term, common difference and the sum of first 20 terms?
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The third term of an A.P is 7 and the seventh term exceeds three times the third term by 2. Find the first term, common difference and the sum of first 20 terms? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The third term of an A.P is 7 and the seventh term exceeds three times the third term by 2. Find the first term, common difference and the sum of first 20 terms? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The third term of an A.P is 7 and the seventh term exceeds three times the third term by 2. Find the first term, common difference and the sum of first 20 terms?.
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