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If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP?
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If the third and the fourth terms of an AP are increased by 3 and 8 re...
Let the first term of the AP be 'a', and the common difference be 'd'.
Then, the third and fourth terms of the AP are:
3rd term = a + 2d
4th term = a + 3d
If we increase the third and fourth terms by 3 and 8 respectively, we get:
3rd term + 3 = a + 2d + 3
4th term + 8 = a + 3d + 8
The new sequence formed by the first four terms of the AP is an arithmetic progression with first term 'a+3' and common difference 'd+5'.
If this sequence is a geometric progression, then its common ratio must be equal to the ratio of its fourth term to its third term:
(a+3+3d+15)/(a+3+2d+3) = (a+3d+8)/(a+2d+3)
Solving this equation for 'a', we get:
a = -15
The sum of the first four terms of the AP is:
a + (a + d) + (a + 2d) + (a + 3d) = -15 + (-15 + d) + (-15 + 2d) + (-15 + 3d) = -30 + 6d
Therefore, the sum of the first four terms of the AP is -30 + 6d.
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Most Upvoted Answer
If the third and the fourth terms of an AP are increased by 3 and 8 re...
Given Information:
- The third and fourth terms of an AP are increased by 3 and 8 respectively.
- The first four terms form a GP.
Let's Solve the Problem:
Step 1: Assume the AP
Let the first term of the AP be 'a' and the common difference be 'd'.
So, the fourth term of the AP is a + 3d, and the fifth term is a + 4d.
Step 2: Form a GP
According to the given information, if we increase the third term by 3 and the fourth term by 8, the terms form a GP.
So, the GP formed is a, a + d, a + 3d, a + 11d.
Step 3: Find the Common Ratio of the GP
The common ratio of a GP is found by dividing any term by the previous term.
Hence, (a + 3d) / (a + d) = (a + 11d) / (a + 3d).
Solving this equation, we get d = 2a.
Step 4: Find the Sum of the First Four Terms
The sum of the first four terms of an AP is given by (4/2) * [2a + 3d] = 2[2a + 6a] = 16a.
Therefore, the sum of the first four terms of the AP is 16a.
- The third and fourth terms of an AP are increased by 3 and 8 respectively.
- The first four terms form a GP.
Let's Solve the Problem:
Step 1: Assume the AP
Let the first term of the AP be 'a' and the common difference be 'd'.
So, the fourth term of the AP is a + 3d, and the fifth term is a + 4d.
Step 2: Form a GP
According to the given information, if we increase the third term by 3 and the fourth term by 8, the terms form a GP.
So, the GP formed is a, a + d, a + 3d, a + 11d.
Step 3: Find the Common Ratio of the GP
The common ratio of a GP is found by dividing any term by the previous term.
Hence, (a + 3d) / (a + d) = (a + 11d) / (a + 3d).
Solving this equation, we get d = 2a.
Step 4: Find the Sum of the First Four Terms
The sum of the first four terms of an AP is given by (4/2) * [2a + 3d] = 2[2a + 6a] = 16a.
Therefore, the sum of the first four terms of the AP is 16a.
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If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP?
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If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP?.
If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the third and the fourth terms of an AP are increased by 3 and 8 respectively, then the first four terms form a GP,find the sum of first four terms of AP?.
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