Arithmetic Progressions Set-2

Q1. In an AP, $a_5=20$ and $a_{12}=-1$. Find the first term $a_1$ and common difference $d$.

Q2. Three numbers are in AP. Their sum is $15$ and the sum of their squares is $83$. Find the three numbers (in increasing order).

Q3. For an AP it is given that the sum of the first $7$ terms is $49$ and the sum of the next $7$ terms (terms $8$ to $14$) is $84$. Find the first term $a$ and common difference $d$.

Q4. Consider the AP $5,\,2,\,-1,\ldots$ (common difference $d=-3$). For which positive integers $n$ is the sum of the first $n$ terms $S_n$ positive?

Q5. An AP has $21$ terms. The sum of the terms in odd positions (1st, 3rd, 5th, ..., 21st) is $110$ and the sum of the terms in even positions (2nd, 4th, ..., 20th) is $100$. What is the middle term (11th term) of the AP?

Q6. The sum of the first $15$ terms of an AP is $345$ and its common difference is $3$. What is the first term of the AP?

Q7. In an AP, the $5$th term is $3$ and the $12$th term is $24$. What is the sum of the first $20$ terms?

Q8. An AP has $20$ terms. The sum of the $10$ odd-positioned terms (1st, 3rd, ..., 19th) is $220$ and the sum of the $10$ even-positioned terms (2nd, 4th, ..., 20th) is $200$. What are the first term $a$ and common difference $d$ of the AP?

Q9. For an AP, the sum of the first $10$ terms exceeds the sum of the next $10$ terms by $10$. What is the common difference $d$ of the AP?

Q10. Let $S_n$ denote the sum of the first $n$ terms of an AP. If $S_n=n^2$ for every positive integer $n$, what is the $10$th term of this AP?

Q11. The 4th term of an AP is $13$ and the 9th term is $28$. What is the 7th term?

Q12. In an AP, the sum of the first $20$ terms is $2100$ and the sum of the first $10$ terms is $650$. The $15$th term of the AP is:

Q13. For an AP, $t_7+t_{13}=52$ and $t_{13}-t_7=18$. The sum of the first $8$ terms, $S_8$, equals:

Q14. In an AP, the sum of the first $6$ terms equals the sum of the next $9$ terms. The ratio $a/d$ (first term to common difference) is:

Q15. The sum of the first $4$ terms of an AP is $40$ and they satisfy $t_1\cdot t_4 = t_2\cdot t_3$. The four terms are: