the sum of four consecutive numbers in an Ap is 37 and the ratio of product of first and last term to product of middle is 7:15. find the numbers.
Ritam1111:
this question was stolen from cbse board exam 2018
Aare par what is the answer of the question
a =8
d = + and - 2
same question came in board exam
Answers
Answered by
138
Let (a - 3d) , (a - d) , (a + d) and (a + 3d) are four consecutive terms in AP.
Now, A/C to question,
Sum of four consecutive terms = 37
(a - 3d) + (a - d) + (a + d) + (a + 3d) = 37
4a = 37 ⇒a = 37/4
Now, question again said , ratio of products of first and last term to the product of middle is 7 : 15
e.g., (a -3d)(a + 3d)/(a - d)(a + d) = 7/15
15(a² - 9d²) = 7(a² - d²)
15a² - 135d² = 7a² - 7d²
15a² - 7a² = 135d² - 7d²
8a² = 128d² ⇒a² = 16d²
Taking square root both sides,
a = ± 4d
so, d = ±a/4
so, d = ± 37/16
Now, numbers are :
(a - 3d) = 37/4 - 3×37/16 = 37/16
(a - d) = 37/4 - 37/16 = 111/16
(a + d) = 37/4 + 37/16 = 185/16
(a + d) = 37/16 + 3 ×37/16 = 259/16
Now, A/C to question,
Sum of four consecutive terms = 37
(a - 3d) + (a - d) + (a + d) + (a + 3d) = 37
4a = 37 ⇒a = 37/4
Now, question again said , ratio of products of first and last term to the product of middle is 7 : 15
e.g., (a -3d)(a + 3d)/(a - d)(a + d) = 7/15
15(a² - 9d²) = 7(a² - d²)
15a² - 135d² = 7a² - 7d²
15a² - 7a² = 135d² - 7d²
8a² = 128d² ⇒a² = 16d²
Taking square root both sides,
a = ± 4d
so, d = ±a/4
so, d = ± 37/16
Now, numbers are :
(a - 3d) = 37/4 - 3×37/16 = 37/16
(a - d) = 37/4 - 37/16 = 111/16
(a + d) = 37/4 + 37/16 = 185/16
(a + d) = 37/16 + 3 ×37/16 = 259/16
yes see in net
it is told that there are 4 consecutive terms,i.e,a,a+d,a+2d,a+3d. and then the answer will be 2,6,10,14.
it is not necessary that all terms will be +ve
yes it is. because there sum is 32.
The no. which are given in the answer, you can add them and check wether the answer is 32 or not
please explain me I don't understand it
let the no. be a,a+d,a+2d,a+3d. A/Q, a+a+d+a+2d+a+3d=32, then the equation formed will be 2a+3d=32--1eq. then,a(a+3d)/(a+d)(a+2d)=7/15. solving this equation,we get,4a^2+6ad+7d^2=0--2nd eq., then square both the sides of eq.1,we get, 4a^2+9d^2+12ad=144--eq.3. then solve both the eq.2 and 3, then we get a=2 and d=4. then the answer will be 2,6,10,14. you can also verify this answer.
this was asked in boards
yes
thanks
Answered by
114
Hey there!
As we are talking about 4 consecutive terms, We consider them to be a - 3d , a - d, a +d, a + 3d [ Selection also stay as a factor for proper solution ]
{ When we talk about 3 terms ,We take a-d, a, a+d }
Now Given,
Sum of four consecutive terms = 37 .
➢ (a-3d)+(a-d) + ( a +d) + ( a + 3d ) = 37
➢ 4a = 37
➢ a = 37/4 .
Also, Ratio of products last terms to the products of ones in the middle is given as 7 : 15 .
➢( a - 3d) ( a + 3d) / (a-d) (a+d) = 7/15
➢ (a²-9d²) / a²-d² = 7/15
➢ 15 ( a² - 9d² ) = 7 ( a² - d² )
➢ 15a² - 135d² = 7a² - 7d²
➢ a²(15-7) = d²(135-7)
➢ a²(8) = d²(128)
➢ a²/d² = 128/8
➢a²/d² = 64/4
➢ a/d = √(64/4) = 8/2 = 4/1
Now,
➢ 1 ( a) = 4(d) [ We have taken 4 as it is principal square root, taking " -4 " would also give the same answers ]
We already have found the value of a as 37/4
➢ 37/4 = 4d
➢ 37/16 = d.
Now,
The numbers are,
a-3d = (4d - 3d) = d = 37/16
a - d = (4d - d) = 3d = 111/16
a+d = (4d+d) = 5d = 185/16
a+3d = (4d + 3d) = 7d = 7(37)/16 = 259/16
As we are talking about 4 consecutive terms, We consider them to be a - 3d , a - d, a +d, a + 3d [ Selection also stay as a factor for proper solution ]
{ When we talk about 3 terms ,We take a-d, a, a+d }
Now Given,
Sum of four consecutive terms = 37 .
➢ (a-3d)+(a-d) + ( a +d) + ( a + 3d ) = 37
➢ 4a = 37
➢ a = 37/4 .
Also, Ratio of products last terms to the products of ones in the middle is given as 7 : 15 .
➢( a - 3d) ( a + 3d) / (a-d) (a+d) = 7/15
➢ (a²-9d²) / a²-d² = 7/15
➢ 15 ( a² - 9d² ) = 7 ( a² - d² )
➢ 15a² - 135d² = 7a² - 7d²
➢ a²(15-7) = d²(135-7)
➢ a²(8) = d²(128)
➢ a²/d² = 128/8
➢a²/d² = 64/4
➢ a/d = √(64/4) = 8/2 = 4/1
Now,
➢ 1 ( a) = 4(d) [ We have taken 4 as it is principal square root, taking " -4 " would also give the same answers ]
We already have found the value of a as 37/4
➢ 37/4 = 4d
➢ 37/16 = d.
Now,
The numbers are,
a-3d = (4d - 3d) = d = 37/16
a - d = (4d - d) = 3d = 111/16
a+d = (4d+d) = 5d = 185/16
a+3d = (4d + 3d) = 7d = 7(37)/16 = 259/16
ok
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