Let an be the nth term of an AP- If -100r-1a2r- and -100r-1a2r-1- then common difference of an AP is -

The correct option is B α β100 →∑_r=1^100a_2r=α⇒ a_2+a_4+...+a_200=α ∑_r=1^100 a_2r 1=β⇒ a_1 +a_3+...+a_199=β (a_2 a_1)+(a_4 a_3)+....+(a ...

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Byju's Answer

Standard XII

Mathematics

Sum of n Terms

Let an be t...

Question

Let $a_{n}$ be the $n^{t h}$ term of an $A P$ . If $100 \sum r = 1 a_{2 r} = α a n d 100 \sum r = 1 a_{2 r - 1} = β$ then common difference of an $A P$ is -

\frac{α - β}{100}

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α - β

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\frac{α - β}{200}

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β - α

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Solution

The correct option is B $\frac{α - β}{100}$
$\to 100 \sum r = 1 a_{2 r} = α \Rightarrow a_{2} + a_{4} + . . . + a_{200} = α$
$100 \sum r = 1 a_{2 r - 1} = β \Rightarrow a_{1} + a_{3} + . . . + a_{199} = β$
$(a_{2} - a_{1}) + (a_{4} - a_{3}) + . . . . + (a_{200} - a_{99}) = α - β$
$\Rightarrow d + d + . . . . d = α - β$
$\Rightarrow 100 d = α - β$
$\Rightarrow d = \frac{α - β}{100}$

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Similar questions

Let $a_{n}$ be the nth term of an AP. If $Σ_{r = 1}^{50} a_{2 r} = α$ and $Σ_{r = 1}^{50} a_{2 r - 1} = β$ , then the common difference of the A.P

Let $a_{n}$ be the $n^{t h}$ term of an AP. If $Σ_{r = 1}^{50} a_{2 r} = α$ and $Σ_{r = 1}^{50} a_{2 r - 1} = β$ , then what is the common difference of the A.P.

Q. Let

a_{n}

be the the

n^{t h}

term of a G.P. of positive numbers such that

100 \sum n = 1 a_{2 n} = α

and

100 \sum n = 1 a_{2 n - 1} = β, α \neq β .

Then the common ratio of G.P. is

Q. Let a_n be the nth term of the G.P. of positive numbers. Let

\sum_{n = 1}^{100} a_{2 n} = α and \sum_{n = 1}^{100} a_{2 n - 1} = β,

such that α ≠ β. Prove that the common ratio of the G.P. is α/β.

Q. Let

a_{n}

be the

n^{t h}

term of a

G . P .

of positive numbers. Let

100 \sum n = 1 a_{2 n} = α

100 \sum n = 1 a_{2 n - 1} = β

such that

α \neq β

Then the common ration of the

G . P .

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Let an be the nth term of an AP. If 100∑r=1a2r=α and100∑r=1a2r−1=β then common difference of an AP is -

The correct option is B α−β100→100∑r=1a2r=α⇒a2+a4+...+a200=α100∑r=1a2r−1=β⇒a1+a3+...+a199=β(a2−a1)+(a4−a3)+....+(a200−a99)=α−β⇒d+d+....d=α−β⇒100d=α−β⇒d=α−β100

Let $a_{n}$ be the $n^{t h}$ term of an $A P$ . If $100 \sum r = 1 a_{2 r} = α a n d 100 \sum r = 1 a_{2 r - 1} = β$ then common difference of an $A P$ is -