The product of the fourth term and the fifth term of an arithmetic progression is 456. Division of the ninth term by the fourth term of the progression gives quotient as 11 and the remainder as 10. Find the first term of the progression *
Answers
Answer:
Step-by-step explanation:
Given
(a+3d)(a+4d)=456.....(1)
(a+8d)=(a+3d)11+10
=> a+8d=11a+33d+10 => 10a+25d+10=0 => d= -2(1+a)/5
from the given opts nly -56 nd -66 satisfy
=> d = 22 nd 26 for a= -56 nd -66 respectvly
Subst a nd d values in (1)
a=-66 nd d= 26 satisfies
second method:-
let 2, 2 + d and 2+2d are first three terms of arithmetic progression
let 2, 2r and 2r^2 are first three terms of geometric progression.
since third terms are equal
2+2d = 2r^2 ==> d = r^2 - 1 --------(1)
2nd term of AP exceeds 2nd term of GP by 0.25
2 + d = 2r + 0.25 ==> d = 2r - 1.75 -----(2)
from (1) and (2)
r^2 - 1 = 2r - 1.75
r^2 - 2r + 0.75 = 0
multiply with 4
4r^2 - 8r + 3 = 0
=> (2r - 3)(2r - 1) = 0
r = 1/2 or 3/2
substitute in eqn (1)
d = r^2 - 1
d = (1/4) - 1 and (9/4) - 1
d = -3/4 and 5/4
A P terms are 2, 5/4, 1/2, -1/4, - 1 and 2, 13/4, 9/2, 23/4, 7
GP terms are 2, 1, 1/2, 1/4, 1/8 and 2, 3, 9/2, 27/4, 81/8
2)
let the AP terms be be, a - 3d, a - 2d, a - d, a, a + d, a+2d, a+3d, a+4d, a+5d
4 th term of AP = a
5 term = a + d
(a4) × (a5) == > a(a+d) = 456
a^2 + ad = 456 ------------(1)
9 th term = a + 5d
a9/a4 = (a+5d)/a
=> (a+5d) /a = 11 + ( 10 / a )
=> (a + 5d) /a = (11a + 10 ) /a
a + 5d = 11a + 10
10a + 10 = 5d
d = 2(a + 1) ---------(2)
substitute d = 2(a + 1) in eqn (1), a^2 + ad = 456
a^2 + 2a (a+ 1) = 456
=> a^2 + 2a^2 + 2a = 456
=> 3a^2 + 2a - 456 = 0
=> 3a^2 + 38a - 36a - 456 = 0
=> 3a (a - 12) + 38(a - 12) = 0
(3a + 38)(a - 12) = 0
a = -38/3 or 12
substitute in eqn (2)
d =2 ( -38/3 + 1 ) or 2 (12 + 1)
d = -70/3 or 26
first term = a - 3d
= -38/3 + 70 = 172/3 or 12 - 78 = -66
first term = -66 or 172/3
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@GauravSaxena01
Let a be the first term and d be the common difference (CD)
Fourth term = a+3d, fifth term = a+4d.
So we have (a+3d)(a+4d) = 456 ( 1 eq)
Ninth term = a +8d and so a+8d = 11(a+3d)+ 10.
Hence d = -(2+2a)/5.
Substituting in (1) we will get
[a- 3(2+2d)/5][a- 4(2+2d)/5] =456.
Simplifying we get 3a^2 +26a-11352= 0
i.e., 3a^2 +198a-172a -11352=0
(a+66)(3a-172) =0
So a= - 66 or 172/3