NIMCET 2025 Paper

We want to check which number of rows cannot form a decreasing-by-3 sequence whose sum is 630 .
A decreasing-by- 3 sequence for $\mathit{n}$ rows looks like:
$\mathit{a},\left(\mathit{a}-3\right),\left(\mathit{a}-6\right),...,\left(\mathit{a}-3\left(\mathit{n}-1\right)\right)$
Sum of $\mathit{n}$ rows:
$\mathit{S}=\frac{\mathit{n}}{2}\left[2\mathit{a}-3\left(\mathit{n}-1\right)\right]=630$
We test each option to see if it gives a positive integer value of $\mathit{a}$.
Check $\mathit{n}=3$
$\frac{3}{2}\left(2\mathit{a}-6\right)=630$
$2\mathit{a}-6=420\Rightarrow 2\mathit{a}=426\Rightarrow \mathit{a}=213$
✓ Valid (integer)
Check $\mathit{n}=4$
$2\left(2\mathit{a}-9\right)=630$
$2\mathit{a}-9=315\Rightarrow 2\mathit{a}=324\Rightarrow \mathit{a}=162$
✓ Valid (integer)
Check $\mathit{n}=5$
$\frac{5}{2}\left(2\mathit{a}-12\right)=630$
$2\mathit{a}-12=252\Rightarrow 2\mathit{a}=264\Rightarrow \mathit{a}=132$
✓ Valid (integer)
Check $\mathit{n}=6$
$3\left(2\mathit{a}-15\right)=630$
$2\mathit{a}-15=210\Rightarrow 2\mathit{a}=225\Rightarrow \mathit{a}=112.5$
$\mathit{x}$ Not an integer → impossible