If another series follows the same pattern as the given series and the first term of this new series...........

Quantitative Aptitude (SBI-PO) Number series

Direction: Study the following information and answer the questions that follow.
A number series is given as
20, a, b, c, d, 65
Where a, b, c and d are missing terms.
It is also given that:
I. a – 20 = (x² + y)
II. The value of b is greater than a and the difference of b and a is equal to the [(x + 1)² + y].
III. The value of c is [(x + 2)² + y] more than b and the value of d is [(x + 3)² + y] more than c.
Note: x is equal to the HCF of 2 prime numbers and the value of y is equal to the smaller root of the quadratic equation z² – z – 6 = 0.

If another series follows the same pattern as the given series and the first term of this new series is 29, then find the fifth term of this new series.

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Explanation:

x = HCF of 2 prime numbers = 1 (Two prime numbers are always co-prime to each other)
z² – z – 6 = 0
⇒ z²– 3z + 2z – 6 = 0
⇒ z(z – 3) + 2(z – 3) = 0
⇒ (z + 2)(z – 3) = 0
So, roots of z² – z – 6 = 0 are 3 and –2
Thus, y = –2
Given, a – 20 = (x² + y)
a = 20 + (12 – 2) = 20 – 1 = 19
b – a = [(x + 1)² + y] = [(1 + 1)²– 2] = 2
⇒ b = a + 2 = 19 + 2 = 21
Also given,
c = b + [(x + 2)² + y] = 21 + [(1 + 2)²– 2] = 21 + 9 – 2 = 28
And
d = c + [(x + 3)² + y] = 28 + [(1 + 3)²– 2] = 28 + 16 – 2 = 42

Given first term of new series = 29
Second Term = 29 + (1² – 2) = 28
Third Term = 28 + [(1 + 1)² – 2] = 30
Fourth Term = 30 + [(1 + 2)² – 2] = 37
Fifth Term = 37 + [(1 + 3)² – 2] = 51
Hence, option 5 is the correct answer.