In a company's internal performance index, the score recorded on day \(n\) is denoted by \(s_n\). The scores form a sequence of integers such that \(s_1 = 40\) and \(s_n < s_{n-1}\) for every integer \(n > 1\).

From which of the following statements can it be concluded that \(s_{20}\) is positive?

I. \(s_{25} = \frac{s_{24}}{2}\) II. The sum of the first \(42\) scores is positive. III. \(s_{29} > s_{27} - s_{28}\)

App models a student's daily focus-adjustment value using the sequence \(f(n) = 5\left(-\frac{1}{2}\right)^n\). The sequence starts from \(n = 1\).

What are the first two consecutive terms for which \(\left|f(n) - f(n+1)\right| < \frac{1}{1000}\)?

During an MBA college fest, coupon IDs are numbered from \(1\) to \(1000\). A team wants to select coupon IDs that form an arithmetic progression, with the first selected ID as \(1\) and the last selected ID as \(1000\).

How many such arithmetic progressions can be formed if each progression must have at least \(3\) elements?

A sequence \(p_1, p_2, p_3, \ldots\) is defined by the following rules:

I. \(p_1 = 1\) II. \(p_{2n} = n \times p_n\) for every positive integer \(n\)

What is the value of \(p_{2^{100}}\)?

Let \(J = 2 + 4 + 6 + 8 + \cdots + 98 + 100\) and \(K = 1 + 3 + 5 + 7 + \cdots + 97 + 99\).

Then the value of \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \cdots + 97^2 - 98^2 + 99^2 - 100^2\) is:

An infinite sequence \(A\) is defined as \(a_1 = 3\) and \(a_n = a_{n-1} + 3 \times 10^{n-1}\) for all \(n > 1\).

If each term in the product \(x_1 \times x_2 \times x_3 \times \cdots \times x_k\) is chosen from sequence \(A\), repetitions are allowed, and the product equals \(173,446,418,443,770,747\), which of the following could be the value of \(k\)?

\(\frac{1}{a+b}\), \(\frac{1}{b+5}\), and \(\frac{1}{a+5}\) are three terms of an arithmetic progression in the given order.

Which of the following represents three consecutive terms in that arithmetic progression?

The second term of a geometric progression is \(1000\), and the common ratio is \(r = \frac{1}{n}\), where \(n\) is a natural number.

Let \(P(n)\) denote the product of the first \(n\) terms of this geometric progression. If \(P(6) > P(5)\) and \(P(6) > P(7)\), what is the sum of all possible values of \(n\)?

Seat numbers are arranged in consecutive groups as \(\{1\}\), \(\{2, 3\}\), \(\{4, 5, 6\}\), \(\{7, 8, 9, 10\}\), \(\ldots\).

Each group has one more number than the previous group, and the first number of each group is one more than the last number of the previous group.

Let \(S_n\) be the sum of the numbers in the \(n^{\text{th}}\) group. What is the value of \(S_{50}\)?

Mentor takes a sequence of consecutive positive integers. The square of the arithmetic mean of the sequence is equal to the difference between the squares of the least and greatest terms.

If the square of the arithmetic mean is less than \(1000\), how many such sequences are possible?

In a CAT mock-test problem, a sequence of real numbers \(a_1, a_2, \ldots, a_n\) is such that \(a_1 + a_2 + \cdots + a_n = 3n^2 - 5n - 8\) for every positive integer \(n \ge 2\).

Find the value of \(k\) such that \(a_k = 112\).

A student club in a B-school has exactly \(5\) new members at the end of its first week. From the second week onward, each member who joined in the previous week, and only those members, brings exactly \(x\) new members into the club.

If \(y\) is the number of new members brought into the club during the \(12^{\text{th}}\) week, which of the following could be \(y\)?

An infinite integer sequence \(b_1, b_2, b_3, \ldots\) is such that \(b_1 = -17\) and \(b_{n-1} - 7 < b_n < b_{n-1} - 2\) for every \(n > 1\).

If \(b_x = -53\), how many different values can \(x\) take?

A sequence \(A_n\) is defined such that \(A_{n+1} = \frac{n+1}{A_n}\) for every positive integer \(n\).

What is the number of positive factors of the product \(A_1 \times A_2 \times A_3 \times A_4 \times A_5 \times A_6\)?

A sequence \(a_1, a_2, a_3, \ldots\) is defined such that \(a_n - a_{n-1} = 2\). Let \(S_n\) denote the sum of the first \(n\) terms of this sequence, and suppose \(a_3 = 5\).

How many ordered pairs of positive integers \((m,n)\) exist such that \(S_m - S_n = 65\)?

A forest worker in an Indian wildlife reserve was asked to tag trees in one section using numbers starting from \(1\), then \(2\), \(3\), and so on. After finishing the work, he added all the tag numbers and got a total of \(1000\).

Later, he realised that one tree's tag number had been added twice. Which number was counted twice?

An infinite sequence is defined as \(S_1 = 2\), \(S_2 = 22\), \(S_3 = 222\), \(\ldots\), and \(S_k = S_{k-1} + 2(10^{k-1})\).

If \(p\) is the sum of the first \(30\) terms of this sequence, what is the \(11^{\text{th}}\) digit of \(p\), counting from right to left from the units digit?

For every positive integer \(n\), the \(n^{\text{th}}\) term of a sequence used in a CAT Quant practice set is given by \((-2)^{n+2} \times \frac{6}{8^{n-1}}\).

If \(S\) is the sum of the first \(8\) terms of the sequence, which of the following accurately represents \(S\)?

In an Indian housing lane, houses on one side of the road are numbered using consecutive positive even numbers. The sum of the house numbers in that row is \(170\).

If there are at least \(6\) houses in that row and \(a\) is the number of the \(6^{\text{th}}\) house, then which of the following is true?

The value of \(A\) is defined as \(A = \frac{1}{1 \times 2 \times 3} + \frac{1}{2 \times 3 \times 4} + \cdots + \frac{1}{48 \times 49 \times 50}\).

What is the value of \(A\)?

The sum of \(4\) consecutive odd integers is equal to the sum of \(3\) consecutive even integers.

Given that the middle term of the even integers is greater than \(101\) and less than \(200\), how many such sets of integers can be formed?

Three real numbers \(a\), \(b\), and \(c\) are consecutive terms of a geometric progression. It is given that \(\left|a+b+c\right| = 15\), the median of these three numbers is \(a\), and \(b = 10\).

If \(a > c\), what is the product of the first \(4\) terms of this geometric progression?

Asequence \(t_1, t_2, t_3, \ldots\) is such that for every positive integer \(n \ge 2\), the sum of the first \(n\) terms is given by \(t_1 + t_2 + \cdots + t_n = 2n^2 + 9n + 13\).

If \(t_k = 103\), what is the value of \(k\)?

Asequence of numbers \(u_1, u_2, u_3, \ldots\) is defined as follows:

\(u_{2n} = u_2 \times u_n + 1\) \(u_{2n+1} = u_2 \times u_n - 2\)

If \(u_7 = 2\) and \(0 < u_1 < 1\), what is the value of \(u_{25}\)?

A student club in a B-school has exactly \(5\) new members at the end of its first week. From the second week onward, each member who joined in the previous week, and only those members, brings exactly \(x\) new members into the club.

If \(y\) is the number of new members brought into the club during the \(12^{\text{th}}\) week, which of the following could be \(y\)?