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Arithmetic Progression

If tn be th...

Question

If $t_{n}$ be the $n^{t h}$ term of the series 1 + 3 + 7 + 15 + ............. then

t_{5} + 1 = 32

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t_{7} = 2^{7} + 1

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t_{10} = 2^{10} - 1

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t_{100} = 2^{50} + 1

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Solution

The correct options are
A $t_{5} + 1 = 32$
C $t_{10} = 2^{10} - 1$
$1 + 3 + 7 + 15 + . . . . = 1 + (1 + 2) + (3 + 2^{2}) + (7 + 2^{3})$
$= 1 + (1 + 2) + (1 + 2 + 2^{2}) + (1 + 2 + 2^{2} + 2^{3})$
$\Rightarrow t_{n} = 1 + 2 + 2^{2} + 2^{3} + . . . . + 2^{n - 1}$

$\Rightarrow t_{n} = \frac{2^{n} - 1}{2 - 1} = 2^{n} - 1$

$∴ t_{5} = 2^{5} - 1 = 32 - 1 \Rightarrow t_{5} + 1 = 32$
$t_{10} = 2^{10} - 1$

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