Let the positive integers be written in the form: If the kth row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is___________

Engineering

Mathematics

Introduction to Logarithm

Question

Let the positive integers be written in the form:

If the kth row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is____________ .

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Last term of each row follows sequence 1, 3, 6, 10, 15, ......
Differences 2, 3, 4, 5, ..... which are in A.P.
t_n = an² + bn + c
1 = a + b + c, 3 = 4a + 2b + c, 6 = 9a + 3b + c

$\Rightarrow 2 = 3 a + b, 3 = 5 a + b \Rightarrow 1 = 2 a, b = \frac{1}{2}, c = 0$

Last term of nth group $= \frac{1}{2} n^{2} + \frac{1}{2} n$

⇒ 1^st term of nth group $= \frac{1}{2} (n - 1)^{2} + \frac{1}{2} (n - 1) + 1$

$\Rightarrow \frac{n^{2} - n + 2}{2} \leq 5310 \leq \frac{n (n + 1)}{2}$ By trail, $n = 100 \Rightarrow \frac{n (n + 1)}{2} = 5050$

$n = 102 \Rightarrow \frac{n (n + 1)}{2} = 5253, n = 103 \Rightarrow \frac{n (n + 1)}{2} = 5356$

⇒ 5310 lies in 103^rd group.

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