Question

the first term of two A.P's are equal and the ratios of their common difference is 1:2 . If the 7th term of first A.P and 21st term of second A.P are 23 and 125 respectively. Find two A.P's​

Answer 1

Answer:

FIRST A.P = 5,8,11......

SECOND A.P = 5,11,17.....

Step-by-step explanation:

FOR THE FIRST A.P

First term = a

Common difference = d

a7 = 23

a + 6d = 23

a = 23 - 6d ----------------- 1

FOR THE SECOND A.P

First term = a ( given)

common difference = b

a21 = 125

a + 20b = 125 -------------- 2

Replacing the value of a from equation 1 to equation 2.

we get,

a + 20b = 125

23 - 6d + 20b = 125

20b - 6d = 125 - 23

20b - 6d = 102

2(10b - 3d) = 102

10b - 3d = 51 ------------ 3

Given,

d/b = 1/2

let,

d= 1x { For x is some common positive integer}

b = 2x { for x is some common positive integer}

Replacing the value of d and b in equation 3

we get,

10b - 3d = 51

10(2x) - 3(x) = 51

20x - 3x = 51

17x = 51

x = 3

Therefore,

d = 3

b = 6

Replacing the value of d in equation 1

we get,

a + 6d = 23

a = 23 - 6(3)

a = 23 - 18

a = 5

THE FIRST A.P

5,8,11.........

THE SECOND A.P

5,11,17...........

GOOD DAY BRO ;)

Answer 2

$Given:-$

• $find \: \: term \: \: of \: \: two \: \: Ap's \: \: are \: \: equal$

• $the \: \: ratio \: \: the \: \: common \: \: difference \: = \: 1 : 2$

• $7th \: \: term \: \: of \: \: 1st \: \: Ap \: = \: 23$

• $21th \: \: term \: \: of \: \: 2nd \: \: Ap \: = \: 125$

$To \: \: find:-$

• $Both \: \: Ap's$

$Solutions:-$

• $Let \: \: the \: \: common \: \: difference \: \: of \: \: 1st \: \: Ap \: \: be \: \: 1d$

• $Let \: \: the \: \: common \: \: difference \: \: of \: \: 2nd \: \: Ap \: \: be \: \: 2d$

$≫≫ a_n \: = \: a + (n - 1)d$

$⇢ a_7 \: = \: a + 6d$

$⇢ a + 6d \: = \: 23$......(1).

$⇢ a_21 \: = \: a + 20 × 2d$

$⇢ a + 40d \: = \: 125$......(2).

• $Solving \: \: Eq. \: (1) \: \: and \:\: (2).$

$a + 6d \: = \: 23$

$a + 40d \: = \: 125$

$- \: \: \: \: \: - \: \: \: \: \: \: = \: \: \: \: -$

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$⇢ - 34d \: = \: - 102$

$⇢ d \: = \: \frac{-102}{-34}$

$⇢ d \: = \: 3$

• $putting \: \: value \: \: of \: \: d \: = \: 3 \: \: in \: \: Eq. \: (1).$

$⇢ a + 6d \: = \: 23$

$⇢ a + 6 \: × \: 3 \: = \: 23$

$⇢ a + 18 \: = \: 23$

$⇢ a \: = \: 23 \: - \: 18$

$⇢ a \: = \: 5$

• $So, \: \: common \: \: difference \: \: of \: \: 1st \: \: Ap \: \: 1d \: = \: 3$
• $common \: \: difference \: \: of \: \: 2nd \: \: Ap \: \: 1d \: = \: 5$

• $So, \: \: the \: \: Ap's \: \: are:-$

$1st \: \: term \: \: of \: \: 1st \: \: Ap \: = \: 5$

$2nd \: \: term \: \: of \: \: 1st \: \: Ap \: = \: a \: + \: d$

$⇢ 5 \: + \: 3$

$⇢ 8$

$3rd \: \: term \: \: of \: \: 1st \: \: Ap \: = \: a \: + \: 2d$

$⇢ 5 \: + \: 2 \: × \: 3$

$⇢ 5 \: + \: 6$

$⇢ 11$

• $So, \: \: 1st \: \: Ap \: = \: 5, \: 8, \: 11....$

$1st \: \: term \: \: of \: \: 2nd \: \: Ap \: = \: 5$

$2nd \: \: term \: \: of \: \: 2nd \: \: Ap \: = \: a \: + \: d$

$⇢ 5 \: + \: 6$

$⇢ 11$

$3rd \: \: term \: \: of \: \: 2nd \: \: Ap \: = \: a \: + \: 2d$

$⇢ 5 \: + \: 2 \: × \: 6$

$⇢ 5 \: + \: 12$

$⇢ 17$

• $So, \: \: 2nd \: \: Ap \: = \: 5, \: 11, \: 17....$

$≫≫ Two \: \: Ap's \: \: are \: \: as:-$

• $1st \: \: Ap \: = \: 5, \: 8, \: 11$
• $2nd \: \: Ap \: = \: 5, \: 11, \: 17$

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