First day income is 1 dollars . On second day income is 4 dollars. And on third day income is 10 dollars. What is the total income after 30 days? First day income is $1$ dollars . On second day income is $4$ dollars. And on third day income is $10$ dollars. What is the total income after $30$ days ? The answer is given $1335$. But how can I figure a pattern here ? Is it $$1,1+3,1+3+6 ,1+3+6+10 ?$$ Any ideas ?

As others have mentioned, there are many different possibilities for how the sequence is to be reckoned.

If the sequence is like this: $1, 1 + 3*1, 1+3*1 + 3*2, 1+3*1+ 3*2 + 3*3, ...$

then the $n$th term will be $T_n = 1 + \frac 32n(n-1)$ based on the AP sum, properly adjusted.

If the sequence is like this: $1, 1 + 3*2^0, 1+3*2^0 + 3*2^1, 1+3*2^0 + 3*2^1 + 3*2^2, ...$

then the $n$th term will be $T_n = 1 + 3(2^{n-1} - 1)$, based on the GP sum, properly adjusted.

If (and this possibility hasn't yet been mentioned - but it corresponds to what you guessed in your question text, the sequence is like this: $1, 1+(1+2), 1+(1+2)+(1+2+3), 1+(1+2)+(1+2+3)+(1+2+3+4)+...$,

then the $n$th term will be $T_n = \frac 16n(n+1)(n+2)$, a derivation based on the sum of the first $n$ triangular numbers.

The thing to take home from all this is that a sequence must be more rigorously defined before we can work on it.