Let `a_(n)` be the nth term of an AP, if `sum_(r=1)^(100)a_(2r)=alpha " and "sum_(r=1)^(100)a_(2r-1)=beta`, then the common difference of the AP is

Question

Let `a_(n)` be the nth term of an AP, if `sum_(r=1)^(100)a_(2r)=alpha " and "sum_(r=1)^(100)a_(2r-1)=beta`, then the common difference of the AP is

Let `a_(n)` be the nth term of an AP, if `sum_(r=1)^(100)a_(2r)=alpha " and "sum_(r=1)^(100)a_(2r-1)=beta`, then the common difference of the AP is
A. `alpha-beta`
B. `beta-alpha`
C. `(alpha-beta)/(2)`
D. None of these

Monjur Alahi

6 views

2025-01-04 04:42

1 Answers

Related Questions

`|(cos alpha cos beta, cos alpha sin beta,-sin alpha),(-sin beta, cos beta, 0),(sin alpha cos beta,sin alpha sin beta, cos alpha)|` का मान ज्ञात कीजिए

2 Answers 5 Views

सारणिकों के गुणधर्मों का प्रयोग करके निम्नलिखित प्रश्न को सिद्ध कीजिए। `|(alpha, alpha^(2),beta+gamma),(beta, beta^(2),gamma+alpha)|=(beta-gamma)(gamm

2 Answers 5 Views

सारणिकों के गुणधर्मों का प्रयोग करके निम्नलिखित प्रश्न को सिद्ध कीजिए। `|(sin alpha, cos alpha, cos(alpha+delta)),(sin beta, cos beta, cos(beta+delta)

2 Answers 5 Views

सारणिकों के गुणधर्मों के प्रयोग से सिद्ध कीजिए कि - `|(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta)|=(alpha - beta )

2 Answers 5 Views

If `alpha,beta` are roots of `x^2-px+q=0` then find the quadratic equation whose roots are `((alpha^2-beta^2)(alpha^3-beta^3))` and `alpha^2beta^3+alp

2 Answers 5 Views

In the quadratic equation `ax^2 + bx + c = 0`. if `delta = b^2-4ac` and `alpha+beta , alpha^2+beta^2 , alpha^3+beta^3` and `alpha,beta` are the roots

2 Answers 4 Views

Find the coefficient of `alpha^(6)` in the product `(1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3))` `(1+alpha)(1+alpha)(1+alpha)`

2 Answers 4 Views

सिद्ध कीजिए कि - `cos^(-1)((cos alpha+cos beta)/(1+cos alpha cos beta))=2tan^(-1)(tan.(alpha)/(2)tan.(beta)/(2)).`

2 Answers 6 Views

दर्शाइए कि आव्यूह `[(cos alpha, sin alpha, 0),(-sin alpha, cos alpha, 0),(0,0,1)]` का व्युत्क्रम `[(cos alpha, - sin alpha,0),(sin alpha, cos alpha, 0

2 Answers 5 Views

If `A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta, sin 2beta] , [sin 2 beta, -cos2beta]]` where `0 lt beta lt pi/2` then prove th

2 Answers 5 Views

Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - D
Given that, `sum_(r=1)^(100)a_(2r)=alpha`
` implies a_(2)+a_(4)+"....."+a_(200)=alpha" " "....(i)"`
and `sum_(r=1)^(100)a_(2r-1)=beta`
`implies a_(1)+a_(3)+"......"+a_(199)=beta " " "………(ii)"`
On subtracting Eq.(ii) from Eq.(i), we get
` (a_(2)-a_(1))+(a_(4)-a_(3))+"......."+(a_(200)-a_(199))=alpha -beta`
`d+d+"......"` upto 100 terms `=alpha -beta`
[beacause `a_(n)` be the nth term of AP with common difference d]
`100d=alpha-beta`
`d=(alpha-beta)/(100)`.