If S n - a 1- a 2- a 3- a n- where a 1-1 and a n -2 a n -1-1- for n - 2- then the value S 50 isA- 251-52B- 250-52- 250-50D- 251-50

The correct option is A 2^51 52a_1 = 1 a_2 = 2× 1 +1 = 3 a_3 = 2× 3 +1 = 7 Now, S_n=a_1+a_2+a_3+⋯+a_n =1+3+7+⋯+a_n =(2 1)+(2^2 1)+(2^3 1)+⋯+(2^n 1) ...

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Byju's Answer

Standard XII

Mathematics

Sum of n Terms

If S n = a 1+...

Question

If $S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n},$ where $a_{1} = 1$ and $a_{n} = 2 a_{n - 1} + 1,$ for $n \geq 2$ , then the value $S_{50}$ is

2^{51} - 52

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2^{51} - 50

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2^{50} - 50

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2^{50} - 52

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Solution

The correct option is A $2^{51} - 52$
$a_{1} = 1 a_{2} = 2 \times 1 + 1 = 3 a_{3} = 2 \times 3 + 1 = 7$
Now,
$S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n} = 1 + 3 + 7 + \dots + a_{n} = (2 - 1) + (2^{2} - 1) + (2^{3} - 1) + \dots + (2^{n} - 1) = (2 + 2^{2} + 2^{3} + \dots + 2^{n}) - n = 2 (\frac{2^{n} - 1}{2 - 1}) - n = 2^{n + 1} - n - 2$
$∴ S_{50} = 2^{51} - 52$

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Similar questions

Q. If

S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n},

where

a_{1} = 1

and

a_{n} = 2 a_{n - 1} + 1,

for

n \geq 2

, then the value

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Q. In an A.P. a₁, a₂, a₃, ..., a_n, ...., if a₁ = 1, a_n= 20 and S_n= 399, then the value of n is __________.

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If Sn=a1+a2+a3+⋯+an, where a1=1 and an=2an−1+1, for n≥2, then the value S50 is

The correct option is A 251−52a1=1a2=2×1+1=3a3=2×3+1=7 Now, Sn=a1+a2+a3+⋯+an =1+3+7+⋯+an =(2−1)+(22−1)+(23−1)+⋯+(2n−1) =(2+22+23+⋯+2n)−n =2(2n−12−1)−n =2n+1−n−2 ∴S50=251−52

If $S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n},$ where $a_{1} = 1$ and $a_{n} = 2 a_{n - 1} + 1,$ for $n \geq 2$ , then the value $S_{50}$ is