the sum of four consecutive numbers in an AP is 36 and ratio of products of first and last terms to the product of middle term is 45:77, first term of AP can be,product of middle trms is, last term of the AP is, difference between middle term is
Answers
Answer:
Step -1: Finding the required numbers.
Let the four consecutive numbers in A.P. be a−3d,a−d,a+d,a+3d
where a is the first term and d is the common difference of the A.P.
According to the given question,
a−3d+a−d+a+d+a+3d=32
⇒4a=32
⇒a=8…(i)
and
(a−d)(a+d)
(a−3d)(a+3d)
=
15
7
⇒
(a
2
−d
2
)
(a
2
−(3d)
2
)
=
15
7
[∵(a+b)(a−b)=a
2
−b
2
]
⇒15(a
2
−9d
2
)=7(a
2
−d
2
)
⇒15a
2
−135d
2
=7a
2
−7d
2
⇒8a
2
=128d
2
⇒8(8)
2
=128d
2
[From(i)]
⇒d
2
=
128
8×8×8
⇒d
2
=4
⇒d=±2
Case 1: When d=2
The numbers are 8−(3×2),8−2,8+2,8+(3×2)
=2,6,10,14
Case 2: When d=−2
The numbers are 8−(3×−2),8−(−2),8+(−2),8+(3×−2)
=14,10,6,2
Hence, the numbers are 2,6,10,14.