IPMAT Indore 2024
Algebra
Progression & Series
Medium
The sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of for which the sum of its first terms is closest to 100, is
The sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of for which the sum of its first terms is closest to 100, is
✅ Correct Option: 2
Let's break this down so it's crystal clear!We have an infinite geometric progression where: - Sum of infinite GP = - Sum of first terms = - We need to find where sum of first terms is closest to
For an infinite GP: Sum = where So: ... (equation 1)Sum of first terms = ... (equation 2)From equation 2: Substituting in equation 1:
Since the infinite sum exists and is positive, we need . Both satisfy this.If : If : Let's verify both cases satisfy our conditions:Case 1: → First terms: ✓Case 2: → First terms: ✓
Using the formula: Case 1: Case 2:
We want :Case 1: → This is impossible since is always positive!Case 2: → Since is negative when is odd, must be odd.
Testing odd values:: , : , : , : , gives the sum closest to (approximately )Therefore,
For an infinite GP: Sum = where So: ... (equation 1)Sum of first terms = ... (equation 2)From equation 2: Substituting in equation 1:
Since the infinite sum exists and is positive, we need . Both satisfy this.If : If : Let's verify both cases satisfy our conditions:Case 1: → First terms: ✓Case 2: → First terms: ✓
Using the formula: Case 1: Case 2:
We want :Case 1: → This is impossible since is always positive!Case 2: → Since is negative when is odd, must be odd.
Testing odd values:: , : , : , : , gives the sum closest to (approximately )Therefore,
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