Question

find five numbers in AP whose sum is
$12 \frac{1}{2}$
and ratio of first to last is 2:3


​

Answer 1

Answer :

☞ ( 5 / 8 ) , ( 25 / 16 ) , ( 40 / 16 ) , ( 55 / 16 ) , ( 70 / 16 )

Explanation :

● Sum of an AP is given by,

Sn = n / 2 [ 2a + ( n - 1 ) d ]

=> S5 = 5 / 2 [ 2a + ( 5 - 1 ) d ]

=> 25 / 2 = 5 / 2 [ 2a + 4d ]

=> ⋆ a + 2d = 5 / 2 ...... ( 1 st equation )

● Also given, Ratio = 2 : 3

=> a / d = 2 / 3

=> ⋆ a = 2d / 3 ...... ( 2 nd equation )

● Putting values ( 2 ) in ( 1 ) :

=> 2d / 3 + 2d = 5 / 2

=> ( 2 × 2 )d + ( 6 × 2 )d = 5 × 3

=> 4d + 12d = 15

=> 16d = 15

=> ⋆ d = 15 / 16

● Putting ( 2 ) value in ( 1 ) :

=> a = 2d / 3

=> a = 2 / 3 × 15 / 16

=> ⋆ a = 5 / 8

● So, First 5 terms of an AP are,

→ a , ( a + d ) , ( a + 2d ) , ( a + 3d ) , ( a + 4d )

→ ( 5 / 8 ) , ( 5 / 8 + 15 / 16 ) , ( 5 / 8 + ( 2 × 15 / 16 ) ) , ( 5 / 8 + ( 3 × 15 / 16 ) ) , ( 5 / 8 + ( 4 × 15 / 16 ) )

→ ⋆ 5 / 8 , 25 / 16 , 40 / 16 , 55 / 16 , 70 / 16

______________________

[ Eg; Note : Finding 5 terms,

• ( 5 / 8 + 2 × 15 / 16 )

• ( 5 / 8 + ( 2 × 15 / 16 )

• ( 5 / 8 + 30 / 16 ) { LCM = 16 }

• 10 + 30 / 16

• ⋆ 40 / 16

Same for all ]

_______________________

♥ So, It's Done ♥ !!