Question

Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each.

(i) 9, 15, 21, 27, …..

(ii) 11, 6, 1, -4, ……

(iii) √2, √8, √18, √32, ……..

Answer 1

Answer:

The common difference is

1. 9,15,21,27...

3²,3×5,3×7,3×9....

3×3,3×5,3×7,3×9....

Let us take 3,5,7,9 as common and the common difference is 2 the next number in this series is 33 is3×11

2. 11,6,1,-4...

11×1,2×3,1×1,2×-2,.....

Let us take 1,3,1,-2 as common and when the same are multiples of 2 on same side and the next number should be multiple of 11 on both sided.So next number is 11×0 is 0 as when we see in the series the number decreases

3.√2,√8,√18,√32...

√2,√2×4,√2×9,√2×16...

√2,2√2,3√2,4√2...….

We should take 1,2,3,4, as common and in the next series the degree of the radical becomes 5. So the next number is√50 is√2×25 is5√2

Answer 2

Answer:

Step-by-step explanation:

(i) $9, 15, 21, 27, … Here, 15 – 9 = 21 – 15 = 27 – 21 = 6$

(which is constant)

Common difference is 6 Or $d = 6$

Next term = $27 + d = 27 + 6 = 33$

(ii) $11, 6, 1, -4, …$

Here,$6 – 11 = 1 – 6 = -5 -4 – 1 = -5$

(which is constant) d (common difference)

$= -5$

Next term =$-4 – 5 = -9$

(iii$) -1, -5 /6 , -2 /3 , -1 /2 , …………$

$-5/6 – (-1) = 1/6$ and$-2/3 – (-5/6)$

= $1/6 d$(common difference)

= $1/6$

Next term =$-1/2 + 1/6 = -1/3$

(iv)$√2, √8, √18, √32, ……..$

$\sqrt{x} 8 – \sqrt{x} 2 = 2\sqrt{x} 2 – \sqrt{x} 2$

= $\sqrt{x} 2 \sqrt{x} 32 – \sqrt{x} 18 = 4\sqrt{x} 2 – 3\sqrt{x} 2$

= $\sqrt{x} 2 d ($common difference)

=$\sqrt{x} 2$

Next term = $\sqrt{x} 32 + \sqrt{x} 2$

= $4\sqrt{x} 2 + \sqrt{x} 2$

= $5\sqrt{x} 2$

= $\sqrt{x} 50$

$(v) \sqrt{x} 20, \sqrt{x} 45, \sqrt{x} 80, \sqrt{x} 125,$

$\sqrt{x} 20, \sqrt{x} 45, \sqrt{x} 80, \sqrt{x} 125,$

$\sqrt{x} 45 – \sqrt{x} 20$

= $3\sqrt{x} 5 \neq 2\sqrt{x} 5 = \sqrt{x} 5 \sqrt{x} 125 – \sqrt{x} 80$

= $5\sqrt{x} 5 – 4\sqrt{x} 5 = \sqrt{x} 5 d$ (common difference)

= $\sqrt{x} 5$

Next term =$\sqrt{x} 125 + \sqrt{x} 5 = 5\sqrt{x} 5 + \sqrt{x} 5 = 6\sqrt{x} 5 or \sqrt{x} 180$

hope it helps

:)