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If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.?
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If Sn denotes the sum of first n terms of an AP then prove that S12: 3...
Proof: S12 = 3(S8 - S4)
Introduction: The given problem is related to the sum of the first n terms of an AP. In this solution, we will prove that S12 is equal to 3 times the difference between S8 and S4.
Step 1: Find the formula for the sum of n terms of an AP: The sum of the first n terms of an AP can be calculated using the following formula:
S_n = (n/2) [2a + (n-1)d]
where a is the first term of the AP, d is the common difference, and n is the number of terms in the series.
Step 2: Find S12: Using the above formula, we can find S12 as follows:
S12 = (12/2) [2a + (12-1)d]
= 6 [2a + 11d]
Step 3: Find S8 and S4: In order to find the difference between S8 and S4, we need to calculate S8 and S4. Using the same formula, we get:
S8 = (8/2) [2a + (8-1)d]
= 4 [2a + 7d]
S4 = (4/2) [2a + (4-1)d]
= 2 [2a + 3d]
Step 4: Find the difference between S8 and S4: We can now find the difference between S8 and S4 as follows:
S8 - S4 = [4(2a + 7d)] - [2(2a + 3d)]
= 8a + 28d - 4a - 12d
= 4a + 16d
Step 5: Prove that S12 = 3(S8 - S4): We can now substitute the values of S12 and S8 - S4 in the given equation and simplify:
S12 = 6 [2a + 11d]
= 3 [4a + 16d/2] + 3 [2a + 11d/2] - 3 [2a + 3d/2]
= 3 [S8 - S4]
Hence, we have proved that S12 is equal to 3 times the difference between S8 and S4.
Introduction: The given problem is related to the sum of the first n terms of an AP. In this solution, we will prove that S12 is equal to 3 times the difference between S8 and S4.
Step 1: Find the formula for the sum of n terms of an AP: The sum of the first n terms of an AP can be calculated using the following formula:
S_n = (n/2) [2a + (n-1)d]
where a is the first term of the AP, d is the common difference, and n is the number of terms in the series.
Step 2: Find S12: Using the above formula, we can find S12 as follows:
S12 = (12/2) [2a + (12-1)d]
= 6 [2a + 11d]
Step 3: Find S8 and S4: In order to find the difference between S8 and S4, we need to calculate S8 and S4. Using the same formula, we get:
S8 = (8/2) [2a + (8-1)d]
= 4 [2a + 7d]
S4 = (4/2) [2a + (4-1)d]
= 2 [2a + 3d]
Step 4: Find the difference between S8 and S4: We can now find the difference between S8 and S4 as follows:
S8 - S4 = [4(2a + 7d)] - [2(2a + 3d)]
= 8a + 28d - 4a - 12d
= 4a + 16d
Step 5: Prove that S12 = 3(S8 - S4): We can now substitute the values of S12 and S8 - S4 in the given equation and simplify:
S12 = 6 [2a + 11d]
= 3 [4a + 16d/2] + 3 [2a + 11d/2] - 3 [2a + 3d/2]
= 3 [S8 - S4]
Hence, we have proved that S12 is equal to 3 times the difference between S8 and S4.
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If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.? for Class 10 2026 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.? covers all topics & solutions for Class 10 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.?.
If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.? for Class 10 2026 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.? covers all topics & solutions for Class 10 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Sn denotes the sum of first n terms of an AP then prove that S12: 3(S8 - S4)???.?.
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