Arithmetic Progressions Set-1

Q1. If in an AP $a_5=12$ and $a_{12}=33$, what is the first term $a_1$?

Q2. For an AP, it is given that $S_{15}=2S_{10}$, where $S_n$ denotes the sum of first $n$ terms and $S_n=\dfrac{n}{2}\bigl(2a+(n-1)d\bigr)$. What is the value of $\dfrac{a}{d}$?

Q3. The $4$th and $10$th terms of an AP have sum $30$ and product $176$. If $d$ is the common difference, what is $|d|$?

Q4. Assertion (A): An AP has $13$ terms. If the sum of terms at odd positions is $195$ and the sum of terms at even positions is $182$, then the middle term equals $13$.
Reason (R): In any AP with an odd number of terms, the sum of all terms equals the number of terms multiplied by the middle term.

Q5. In an AP, $a_1=5$ and $a_{20}=-30$. How many positive terms are there in this AP?

Q6. An arithmetic progression has first term $a_1=5$ and the sum of the first $10$ terms $S_{10}=200$. Find the common difference $d$.

Q7. In an AP the third term is $10$ and the eighth term is $35$. Find the sum of the first $20$ terms $S_{20}$.

Q8. Let three consecutive terms of an AP be $a,b,c$. If $a+b+c=15$ and $a-2b+3c=4$, determine the ordered triple $(a,b,c)$.

Q9. Assertion (A): If $S_n$, the sum of first $n$ terms of a sequence, is a polynomial in $n$ of degree $2$, then the sequence $\{a_n\}$ is an arithmetic progression.
Reason (R): Because $a_n=S_n-S_{n-1}$ is a polynomial of degree $1$, so consecutive differences of $\{a_n\}$ are constant.

Q10. Can three consecutive terms of a non-constant arithmetic progression form a geometric progression?

Q11. The first term of an AP is $5$ and its $7$th term is $29$. What is the $12$th term?

Q12. The sum of the first $n$ terms of an AP is given by $S_n=3n^2+5n$. Find the common difference of the AP and the value of $S_{10}$.

Q13. In an AP, the $6$th, $10$th and $15$th terms are in geometric progression. If the $20$th term of the AP is $300$, find the first term of the AP.

Q14. Assertion (A): There is no arithmetic progression with non-zero common difference such that its $5$th term is $20\%$ greater than its $2$nd term and its $10$th term is $50\%$ greater than its $4$th term.
Reason (R): If $a$ and $d$ denote the first term and common difference, the two percentage conditions lead to two linear equations in $a$ and $d$ which force $a/d$ to take two different values unless $d=0$, hence no non-zero $d$ satisfies both conditions.

Q15. The sequence $1,2,3,\dots$ is an AP with first term $1$ and common difference $1$. What is the smallest integer $n>1$ such that the sum of the first $n$ terms

is a perfect square?