Test Level 1: Progressions, Sequences & Series - 2 - CAT MCQ

a, ar, ar², ar³
a + ar + ar² + ar³ = 156
a + ar² = 1/5 (ar + ar³)
5a + 5ar² = ar + ar³
5 + 5r² = r + r³
r³ - 5r² + r - 5 = 0
r² (r - 5) + 1(r - 5) = 0
r = 5, r = ±1
a + 5a + 25a + 125a = 156
156a = 156
a = 1

As, 2, a, b, c, d, e, f, 12 are in AP
So, 2 + a + b + c + d + e + f + 12 = 8/2 (2 + 12)
2 + a + b + c + d + e + f + 12 = 4 × (2 + 12)
a + b + c + d + e + f = 4 × (2 + 12) - 2 - 12
a + b + c + d + e + f = 42

Let the AP be a, a + d, a + 2d, …
∵ Tn = a + (n - 1)d
T₃ + T₁₅ = T₆ + T₁₁ + T₁₃
2a + 2d + 14d = 3a + 5d + 10d + 12d
0 = a + 11d
0 = T₁₂
Thus, the 12th term of the AP is 0.

a + (p - 1)d = x … (i)
a + (q - 1)d = y … (ii)
a + (r - 1)d = z … (iii)
Subtracting (ii) from (i), (ii) from (iii) & (iii) from (i), we get
⇒ (p - q)d = x - y

Given: x(q - r) + y(r - p) + z(p - r)

100 + 110 + 120 + ……… + 990
= 10(10 + 11 + …… + 99)

= 10(4950 – 45) = 49,050

t₅ = 2t₃
(a + 4d) = 2(a + 2d)
⇒ a = 0

Since a, b, c are in AP, therefore
b - a = c - b = d and c - a = 2d.
∴ x^{b - c} × y^{c - a} × z^{a - b} = x^-d × y^2d × z^-d
= (xz)^-d × y^2d = y^-2d × y^2d = 1. (∵ y² = xz)

Let the AP be a, a + d, a + 2d, …
∵ T_n = a + (n - 1)d
T₃ + T₁₅ = T₆ + T₁₁ + T₁₃
2a + 2d + 14d = 3a + 5d + 10d + 12d
0 = a + 11d
0 = T₁₂
Thus, the 12th term of the AP is 0.

10 + 12 + 14 + ------- + 200
a = 10, d = 2
T_n = 200 = 10 + (n - 1) × 2
⇒ n = 96


Sum of the numbers, which are multiple of 6:
12 + 18 + _ _ _ _ _ + 198
198 = 12 + (m - 1) 6
m = 32

Required Sum = 10,080 - 3,360 = 6,720