Let an be the n th term of an A-P-If - r-1 100 a 2r - and - r-1 100 a 2r-1 - then the common difference of the A-P is

The correct option is C α β100 ∑_r=1^100 a_2r=α a_2+a_4+a_6+a_8.........+a_200=α ... (i) ∑_r=1^100a_2r 1=β a_1+a_3+a_5+........+a_199=β ...

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Byju's Answer

Standard XII

Mathematics

Definition of Functions

Let an be t...

Question

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

\frac{α - β}{200}

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α - β

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\frac{α - β}{100}

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β - α

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Solution

The correct option is C $\frac{α - β}{100}$
$\sum_{r = 1}^{100} a_{2 r} = α$
$a_{2} + a_{4} + a_{6} + a_{8} . . . . . . . . . + a_{200} = α$ ... (i)
$\sum_{r = 1}^{100} a_{2 r - 1} = β$
$a_{1} + a_{3} + a_{5} + . . . . . . . . + a_{199} = β$ ...(ii)
subtracting (i) and (ii)
$a_{1} - a_{2} + a_{4} - a_{3} + a_{6} - a_{5} + . . . . . . . + a_{200} - a_{199} = α - β$ (terms are in A.P)
$d + d + d + d + . . . . . . d = α - β$
$100 d = α - β$
$a = \frac{α - β}{100}$

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Let an be the nth term of an A.P.If 100∑r=1a2r=α and 100∑r=1a2r−1=β, then the common difference of the A.P is

The correct option is C α−β100∑100r=1a2r=αa2+a4+a6+a8.........+a200=α ... (i)∑100r=1a2r−1=βa1+a3+a5+........+a199=β ...(ii)subtracting (i) and (ii)a1−a2+a4−a3+a6−a5+.......+a200−a199=α−β (terms are in A.P)d+d+d+d+......d=α−β100d=α−βa=α−β100

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