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In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112?
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In an AP., the first term is 2 and sum of first five term is one Fourt...
First term = 2Let d be the common difference of the A.P.Therefore, the A.P. is 2, 2 + d, 2 + 2d, 2 + 3d, …Sum of first five terms = 10 + 10dSum of next five terms = 10 + 35dAccording to the given condition,=>10+10d = 1/4(10+35d)=>40+40d = 10+35d=>30 = −5d=>d = −6∴ a20 = a+(20−1)d = 2+(19)(−6) = 2−114 = −112Hence, the 20th term of the A.P. is –112.
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In an AP., the first term is 2 and sum of first five term is one Fourt...
**Problem Statement:**
We are given an arithmetic progression (AP) where the first term is 2. The sum of the first five terms is one fourth of the sum of the next five terms. We need to prove that the 20th term of this AP is -112.
**Solution:**
Let's solve this problem step by step:
**Step 1: Understanding the given information**
We are given that the first term of the AP is 2. Let's assume the common difference of the AP is 'd'.
**Step 2: Finding the sum of the first five terms**
The sum of the first five terms of an AP can be calculated using the formula:
Sum of first n terms = (n/2) * (2a + (n-1)d)
In this case, n = 5 and a = 2 (first term). Substituting these values in the formula, we get:
Sum of first five terms = (5/2) * (2 + 4d) = 10 + 10d
**Step 3: Finding the sum of the next five terms**
Similarly, the sum of the next five terms of the AP can be calculated using the same formula. In this case, n = 5 and a = 2 + 5d (sixth term). Substituting these values in the formula, we get:
Sum of next five terms = (5/2) * (2 + 5d + 4d) = 10 + 22.5d
**Step 4: Setting up the equation**
According to the given information, the sum of the first five terms is one fourth of the sum of the next five terms. Mathematically, this can be expressed as:
10 + 10d = (1/4) * (10 + 22.5d)
Simplifying this equation, we get:
40 + 40d = 10 + 22.5d
Collecting like terms, we get:
40d - 22.5d = 10 - 40
17.5d = -30
d = -30/17.5
**Step 5: Finding the 20th term**
Now that we have the value of 'd', we can find the 20th term of the AP. Using the formula for the nth term of an AP:
nth term = a + (n-1)d
In this case, a = 2 (first term), n = 20, and d = -30/17.5. Substituting these values in the formula, we get:
20th term = 2 + (20-1) * (-30/17.5) = 2 + 19 * (-30/17.5) = 2 - 570/17.5 = -112
Therefore, the 20th term of the AP is indeed -112.
Hence, we have successfully proven that the 20th term of the given AP is -112.
We are given an arithmetic progression (AP) where the first term is 2. The sum of the first five terms is one fourth of the sum of the next five terms. We need to prove that the 20th term of this AP is -112.
**Solution:**
Let's solve this problem step by step:
**Step 1: Understanding the given information**
We are given that the first term of the AP is 2. Let's assume the common difference of the AP is 'd'.
**Step 2: Finding the sum of the first five terms**
The sum of the first five terms of an AP can be calculated using the formula:
Sum of first n terms = (n/2) * (2a + (n-1)d)
In this case, n = 5 and a = 2 (first term). Substituting these values in the formula, we get:
Sum of first five terms = (5/2) * (2 + 4d) = 10 + 10d
**Step 3: Finding the sum of the next five terms**
Similarly, the sum of the next five terms of the AP can be calculated using the same formula. In this case, n = 5 and a = 2 + 5d (sixth term). Substituting these values in the formula, we get:
Sum of next five terms = (5/2) * (2 + 5d + 4d) = 10 + 22.5d
**Step 4: Setting up the equation**
According to the given information, the sum of the first five terms is one fourth of the sum of the next five terms. Mathematically, this can be expressed as:
10 + 10d = (1/4) * (10 + 22.5d)
Simplifying this equation, we get:
40 + 40d = 10 + 22.5d
Collecting like terms, we get:
40d - 22.5d = 10 - 40
17.5d = -30
d = -30/17.5
**Step 5: Finding the 20th term**
Now that we have the value of 'd', we can find the 20th term of the AP. Using the formula for the nth term of an AP:
nth term = a + (n-1)d
In this case, a = 2 (first term), n = 20, and d = -30/17.5. Substituting these values in the formula, we get:
20th term = 2 + (20-1) * (-30/17.5) = 2 + 19 * (-30/17.5) = 2 - 570/17.5 = -112
Therefore, the 20th term of the AP is indeed -112.
Hence, we have successfully proven that the 20th term of the given AP is -112.
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In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112?
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In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112?.
In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an AP., the first term is 2 and sum of first five term is one Fourth of the next five terms .Show that 20th term is -112?.
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