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A sequence is such that in any set of four consecutive terms, the sum of first and third term is equal to the sum of second and fourth term. If the third term is equal to 5, 26th term is equal to 9 and the sum of first 18 terms is equal to 58, find the first term.
  • a)
    1
  • b)
    2
  • c)
    3
  • d)
    4
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A sequence is such that in any set of four consecutive terms, the sum ...
Let's assume the given sequence is {a, b, 5, d, e, ...} and let's denote the sum of first and third term as S1, and the sum of second and fourth term as S2.

According to the given condition, S1 = S2. Therefore, we can write the equation as:
(a + 5) = (b + d)
We can also write the equation for the sum of the first four terms:
(a + b + 5 + d) = S1 + S2
(a + b + 5 + d) = 2S1

Calculating the sum of the first 18 terms:
S18 = a + (a + b) + 5 + (a + b + 5) + ... + (a + 17(b - a) + 5)
S18 = 18a + 153b

Given that S18 = 58, we can substitute this value into the equation:
18a + 153b = 58

We are also given that the third term is 5, which means d = 5.
Using this information, we can substitute the values into the equation for S1:
(a + 5) = (b + 5)
a = b

Now, we have two equations and three variables (a, b, and S1). We need another equation to solve for a and b.

We are also given that the 26th term is 9. Using the same logic, we can write the equation for the 26th term:
(d + e) = 9
5 + e = 9
e = 4

Now, we have the value of e. We can substitute this value into the equation for S18:
S18 = a + (a + b) + 5 + (a + b + 5) + ... + (a + 17(b - a) + 5)
S18 = 18a + 153b
58 = 18a + 153b

We can solve these two equations simultaneously to find the values of a and b.
18a + 153b = 58
a = b

By solving these equations, we find that a = 1 and b = 1.

Therefore, the first term of the sequence is 1, which corresponds to option A.
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Community Answer
A sequence is such that in any set of four consecutive terms, the sum ...
Let the first term be a, second term be x and the third term be c.
Fourth term = a + c - x
Fifth term = a
Sixth term = x
In this way pattern will repeat.
26th term = 2nd term = x = 9
c = 5
2(a + c)*4 + a + x = 58
9a + 8c + x = 58
9a + 40 + 9 = 58
a = 1
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Given sequence: 3, 11, 34, 69, 131, 223To find the pattern in the sequence, let's calculate the differences between consecutive terms:11 - 3 = 834 - 11 = 2369 - 34 = 35131 - 69 = 62223 - 131 = 92The differences between consecutive terms are not following a consistent pattern. Therefore, the given sequence does not have a simple arithmetic or geometric progression.However, if we observe the differences between the differences, we can see a pattern:23 - 8 = 1535 - 23 = 1262 - 35 = 2792 - 62 = 30The differences between the differences are increasing by a constant value. This indicates that the given sequence might have a polynomial pattern.To find the polynomial pattern, we can use the method of finite differences. Let's calculate the second differences:15 - 12 = 312 - 27 = -1527 - 30 = -3The second differences are not constant. Therefore, the given sequence might have a polynomial of degree 2.Let's assume the polynomial is of the form ax^2 + bx + c. Substituting the given terms into the polynomial, we can form the following equations:1. 3a + b + c = 32. 9a + 3b + c = 113. 16a + 4b + c = 34Solving these equations, we get a = 1, b = 2, and c = 0.Therefore, the polynomial pattern for the given sequence is x^2 + 2x.Now, let's find the missing terms in the sequence

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A sequence is such that in any set of four consecutive terms, the sum of first and third term is equal to the sum of second and fourth term. If the third term is equal to 5, 26th term is equal to 9 and the sum of first 18 terms is equal to 58, find the first term.a)1b)2c)3d)4Correct answer is option 'A'. Can you explain this answer?
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