Test: Sequence & Series - 1 - CAT MCQ

For maximum value of the given sequence to n terms, when the nth term is either zero or the
smallest positive number of the sequence
ie., 50 + (n − 1)(−2) = 0 ⇒ n = 26
Therefore, S₂₆ = 26/2 (50 + 0) = 26 × 25 = 650

Initial height is 30m, after the first bounce, the height reached is 3/4(30). After the second bounce, the height reached = 3*3/4*4(30).

Total distance travelled = 30 + 3/4(30)*2 + 3/4(3/4(30))*2 + …

= 30 + 3/4(30)*2/(1-3/4) = 210m

This is an arithmetic series with first term equal to 82 and common difference equal to 31
Let the number of digits be equal to n.
So,  82 + (n-1) 31 = 609
or, n = 18
So, the sum equals 18/2 ∗(82+609) = 9∗691 = 6219

Let the first term of the AP be a and the common difference be d.
The sixth term of the series will be a + 5d
Given that d should be a factor of a + 5d
=> a + 5d is divisible by d
=> a should be divisible by d
So the required cases are
d = 1, a = 1, 2, 3.......20
d= 2 , a = 2, 4, 6.......14
d = 3, a = 3, 6, 9
d= 4, a = 4
So the required number of AP’s are 20 + 7 + 3 + 1 = 31

Let the first term be a and the common difference be d. We have, (a + 7d)/(a + 16d) = 5/11.
Solving this, we get d = 2a. Product of the first and sixth terms is a (a + 5d) = a(a + 10 a) = 11a² = 176 ⇒ a = + - 4
Since the series has only negative terms, a=-4 and d=-8.
Sum of the first 200 terms = (200/2)(2*-4 + 199*- 8) = -160000

Let the first term be a, second term be x and the third term be c.
Fourth term = a + c - x
Fifth term = a
Sixth term = x
In this way pattern will repeat.
26th term = 2nd term = x = 9
c = 5
2(a + c)*4 + a + x = 58
9a + 8c + x = 58
9a + 40 + 9 = 58
a = 1

The sum of the first 99991 odd numbers is 99991² = 9998200081

Since, Bali’s count from the left goes up by 8, Moti’s count from the right would go down by 8 too. Option (d) is correct.

All the numbers in the series are perfect cubes. 999 is not a cube of any natural number. Hence, Option (d) does not belong to the series.

The series is following the pattern +3, +5, +7, +9, +11, +13 and hence the next term should be 48 + 15 = 63. Answer is option (b).