Arithmetic Progressions

Explanatory Answer

i. 2, \\square), 26

Let the common difference be ‘d’
Let the missing term be x
∴ x = 2 + d ----------(1)
And 26 = x + d --------(2)
Replace x as (2 + d) in equation (2)
26 = 2 + d + d
2d = 24 or d = 12
∴ x = 2 + 12 = 14

ii. -------, 13, -------, 3

4^th term, a₄ is 3
2^nd term, a₂ is 13
a₄ = a₂ + 2d
3 = 13 + 2d
-10 = 2d or d = -5
a₂ = a + d
13 = a – 5 or a = 18
a₃ = a₂ + d
= 13 - 5 = 8
The AP is 18, 13, 8 3

iii. 5, ----, -----, 9½

a = 5, a₄ = \\frac{19}{2})
a₄ = a + 3d
3d = \\frac{19}{2}) – 5 = \\frac{9}{2})
d = \\frac{3}{2})
∴ a₂ = a + d = 5 + \\frac{3}{2}) = \\frac{13}{2})
a₃ = a₂ + d = \\frac{13}{2}) + \\frac{3}{2}) = 8
The AP is 5, 6.5, 8, 9.5

iv. -4, ----, ----, ---, ---, 6

a = -4; a₆ = 6
a₆ = a + 5d
6 = -4 + 5d
5d = 10 or d = 2
a₂ = a + d = -4 + 2 = -2
a₃ = a₂ + d = -2 + 2 = 0
a₄ = a₃ + d = 0 + 2 = 2
a₅ = a₄ + d = 2 + 2 = 4
The AP is -4, -2, 0, 2, 4, 6

v. ----, 38, ---, ----, ----, -22

a₂ = 38, a₆ = -22
a₂ = a + d = 38 --- (1)
a₆ = a + 5d = -22 --- (2)
(2) – (1) => 4d = - 60
d = -15
a₂ = a + d
38 = a - 15
a = 53
a₃ = a₂ + d
= 38 + (-15) = 23
a₄ = a₃ + d
= 23 – 15 = 8
a₅ = a₄ + d = 8 -15 = -7
The AP is 53, 38, 23, 8, -7, -22