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8. sequence and series — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS8. sequence and series1 Mark
MCQ
The value of $\left(2 .{ }^{1} P _{0}-3 .{ }^{2} P _{1}+4 .{ }^{3} P _{2}-\ldots .\right.$ up to $51$ th term)+$\left(1 !-2 !+3 !-\ldots . .\right.$ up to $51^{\text {th }}$ term $)$ is equal to
  • A
    $1+(51) !$
  • B
    $1-51(51) !$
  • ✓
    $1+(52) !$
  • D
    $1$
✓

Answer

Correct option: C.
$1+(52) !$
c
$S =\left(2 \cdot{ }^{1} P _{0}-3 .{ }^{2} p _{1}+4 .{ }^{3} p _{2} \ldots \ldots \ldots\right.$ upto 51 terms $)$

$+(1 !+2 !+3 ! \ldots \ldots \ldots .$ upto 51 terms $)$

$\left[\because{ }^{n} p_{n-1}=n !\right]$

$ \therefore \quad S =$$(2 \times 1 !-3 \times 2 !+4 \times 3 ! \ldots+52.51 !)$

$+(1 !-2 !+3 ! \ldots \ldots \ldots .(51) !) $$=(2 !-3 !+4 ! \ldots \ldots .+52 !) $

$+(1 !-2 !+3 !-4 !+\ldots \ldots+(51) !) $$= 1 !+52 !$

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