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The sequence a1, a2,…an is such that an = an-1 +n*d for all n > 1, where d is a positive integer. If a3 = 20 and a5 = 47, what is the value of a7?
- a)53
- b)65
- c)75
- d)80
- e)86
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
The sequence a1, a2,…an is such that an = an-1 +n*d for all n &#...
Given
- A sequence a1, a2,…an
- an = an-1 +n*d for all n > 1, where d is an integer > 0
- a3 = 20
- a5 = 47
To Find: a7?
Approach
- As an = an-1 +n*d, we can express a7 in terms of a1 and d
- So, we need to find the value of a1 and d.
- As we are given the values of a3 and a5, we will express them in terms of a1 and d to get 2 equations in a1 and d.
- We will then solve these two equations to find out the value of a1 and d.
Working Out
Solving (1) and (2), we have a1 = 5 and d = 3 
Answer: E
Most Upvoted Answer
The sequence a1, a2,…an is such that an = an-1 +n*d for all n &#...
To solve this problem, we need to find the value of a7 in the given sequence. Let's break down the information provided and use it to find the answer.
Given information:
- an = an-1 * n*d for all n ≥ 1
- a3 = 20
- a5 = 47
Finding the common ratio:
To find the common ratio (d) in the sequence, we can use the given information. We know that a3 = 20 and a5 = 47. Using the formula for an, we can write the following equations:
a3 = a2 * 2d
20 = a2 * 2d
a5 = a4 * 4d
47 = a4 * 4d
Dividing the second equation by the first equation, we get:
47/20 = (a4 * 4d) / (a2 * 2d)
47/20 = 2a4/a2
47/20 = 2(a4/a2)
47/20 = 2(a3 * 3d)/(a3 * d)
47/20 = 6d/d
47/20 = 6
47 = 120
Therefore, we have found that the common ratio (d) is 120.
Finding the value of a7:
Now that we know the common ratio (d), we can find the value of a7 using the formula for an:
a7 = a6 * 6d
To find a6, we can use the formula for an again:
a6 = a5 * 5d
a6 = 47 * 5d
Substituting this value into the equation for a7, we get:
a7 = (47 * 5d) * 6d
a7 = 235d * 6d
a7 = 1410d^2
Since d is a positive integer, we can see that the value of a7 is directly proportional to the square of d.
Therefore, to find the value of a7, we need to know the value of d. Unfortunately, the value of d is not provided in the question. Without knowing the value of d, we cannot determine the exact value of a7.
Hence, the given answer options (a, b, c, d, e) are not sufficient to determine the value of a7.
Given information:
- an = an-1 * n*d for all n ≥ 1
- a3 = 20
- a5 = 47
Finding the common ratio:
To find the common ratio (d) in the sequence, we can use the given information. We know that a3 = 20 and a5 = 47. Using the formula for an, we can write the following equations:
a3 = a2 * 2d
20 = a2 * 2d
a5 = a4 * 4d
47 = a4 * 4d
Dividing the second equation by the first equation, we get:
47/20 = (a4 * 4d) / (a2 * 2d)
47/20 = 2a4/a2
47/20 = 2(a4/a2)
47/20 = 2(a3 * 3d)/(a3 * d)
47/20 = 6d/d
47/20 = 6
47 = 120
Therefore, we have found that the common ratio (d) is 120.
Finding the value of a7:
Now that we know the common ratio (d), we can find the value of a7 using the formula for an:
a7 = a6 * 6d
To find a6, we can use the formula for an again:
a6 = a5 * 5d
a6 = 47 * 5d
Substituting this value into the equation for a7, we get:
a7 = (47 * 5d) * 6d
a7 = 235d * 6d
a7 = 1410d^2
Since d is a positive integer, we can see that the value of a7 is directly proportional to the square of d.
Therefore, to find the value of a7, we need to know the value of d. Unfortunately, the value of d is not provided in the question. Without knowing the value of d, we cannot determine the exact value of a7.
Hence, the given answer options (a, b, c, d, e) are not sufficient to determine the value of a7.
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