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If the first term of an arithmetic progression satisfy the equation, 20ds = (/ + 5d)2, where d is the common difference, / is the last term in the sequence and s is the sum of all the n terms. What is the value of the first term of the sequence?
- a)5d
- b)10d
- c)d/5
- d)5/d
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If the first term of an arithmetic progression satisfy the equation, 2...
Let a be the first term of the sequence.
The given equation is true for first term of the A.P.
For n = 1, a = / = s Substituting s = a and / = a in the given equation, we get,
20ad = (a + 5d)2
∴ 20ad = a2 + 10ad + 25d2
∴ a2 - 10ad + 25d2 = 0
∴ (a - 5d)2 = 0
∴ a - 5d = 0
∴ a = 5d
Hence, option 1.
The given equation is true for first term of the A.P.
For n = 1, a = / = s Substituting s = a and / = a in the given equation, we get,
20ad = (a + 5d)2
∴ 20ad = a2 + 10ad + 25d2
∴ a2 - 10ad + 25d2 = 0
∴ (a - 5d)2 = 0
∴ a - 5d = 0
∴ a = 5d
Hence, option 1.
Most Upvoted Answer
If the first term of an arithmetic progression satisfy the equation, 2...
Given information:
- The first term of an arithmetic progression satisfies the equation: 20ds = (/ 5d)^2
- d is the common difference
- / is the last term in the sequence
- s is the sum of all the n terms
To find:
The value of the first term of the sequence
Solution:
Let's assume that the first term of the arithmetic progression is 'a' and the number of terms is 'n'.
Step 1: Finding the last term
The last term of an arithmetic progression (Tn) is given by the formula:
Tn = a + (n-1) * d
In this case, the last term is given as '/'. So we can substitute the values in the formula:
/ = a + (n-1) * d
Step 2: Finding the sum of all the terms
The sum of all the terms of an arithmetic progression (S) is given by the formula:
S = (n/2) * (first term + last term)
Substituting the values, we get:
s = (n/2) * (a + /)
Step 3: Solving the equation
From the given equation, we have:
20ds = (/ 5d)^2
Simplifying the equation, we get:
20ds = /^2 / (25d^2)
Cross-multiplying, we have:
20ds * 25d^2 = /^2
Simplifying further, we get:
500d^3s = /^2
Step 4: Finding the value of 'a'
We can substitute the value of / from Step 1 in the equation from Step 3:
500d^3s = (/^2)
Substituting '/ = a + (n-1) * d':
500d^3s = (a + (n-1) * d)^2
Since we need to find the value of 'a', we can substitute the value of s from Step 2:
500d^3 * [(n/2) * (a + /)] = (a + (n-1) * d)^2
Simplifying the equation, we get:
500d^3 * [(n/2) * (a + a + (n-1) * d)] = (a + (n-1) * d)^2
Simplifying further, we get:
500d^3 * [(n/2) * (2a + (n-1) * d)] = (a + (n-1) * d)^2
Step 5: Finding the value of 'a'
Since the equation is a bit complex, we can use the answer options to simplify the calculations.
Let's consider option A: a = 5d.
Substituting the value of 'a' in the equation, we get:
500d^3 * [(n/2) * (2(5d) + (n-1) * d)] = (5d + (n-1) * d)^2
Simplifying further, we get:
500d^3 * [(n/2) * (10d + (n-1) *
- The first term of an arithmetic progression satisfies the equation: 20ds = (/ 5d)^2
- d is the common difference
- / is the last term in the sequence
- s is the sum of all the n terms
To find:
The value of the first term of the sequence
Solution:
Let's assume that the first term of the arithmetic progression is 'a' and the number of terms is 'n'.
Step 1: Finding the last term
The last term of an arithmetic progression (Tn) is given by the formula:
Tn = a + (n-1) * d
In this case, the last term is given as '/'. So we can substitute the values in the formula:
/ = a + (n-1) * d
Step 2: Finding the sum of all the terms
The sum of all the terms of an arithmetic progression (S) is given by the formula:
S = (n/2) * (first term + last term)
Substituting the values, we get:
s = (n/2) * (a + /)
Step 3: Solving the equation
From the given equation, we have:
20ds = (/ 5d)^2
Simplifying the equation, we get:
20ds = /^2 / (25d^2)
Cross-multiplying, we have:
20ds * 25d^2 = /^2
Simplifying further, we get:
500d^3s = /^2
Step 4: Finding the value of 'a'
We can substitute the value of / from Step 1 in the equation from Step 3:
500d^3s = (/^2)
Substituting '/ = a + (n-1) * d':
500d^3s = (a + (n-1) * d)^2
Since we need to find the value of 'a', we can substitute the value of s from Step 2:
500d^3 * [(n/2) * (a + /)] = (a + (n-1) * d)^2
Simplifying the equation, we get:
500d^3 * [(n/2) * (a + a + (n-1) * d)] = (a + (n-1) * d)^2
Simplifying further, we get:
500d^3 * [(n/2) * (2a + (n-1) * d)] = (a + (n-1) * d)^2
Step 5: Finding the value of 'a'
Since the equation is a bit complex, we can use the answer options to simplify the calculations.
Let's consider option A: a = 5d.
Substituting the value of 'a' in the equation, we get:
500d^3 * [(n/2) * (2(5d) + (n-1) * d)] = (5d + (n-1) * d)^2
Simplifying further, we get:
500d^3 * [(n/2) * (10d + (n-1) *
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