Practice section 6 of 17

Find the sum of the first $20$ terms of the AP $7,13,19,25,\ldots$.

The first seven terms of an AP sum to $28$, and the next eight terms sum to $62$. List the first three terms of the AP.

Alice has $4800$ sweets to divide into bags. She puts $25$ sweets in the first bag and $5$ more in each subsequent bag than in the preceding bag. She stops when there are not enough sweets to fill the next bag in the same manner. Find the number of sweets left behind.

The ninth term of an AP is $43$ and its first $15$ terms sum to $570$. (i) Find its first term $a$ and common difference $d$. (ii) Given $S_n>2265$, find the least possible $n$.

Express $1+4+7+10+\cdots+61$ in sigma notation and find its sum.

Evaluate $\sum_{k=1}^{n}(7k+5)$ for positive integer $n$.