If a1 = 1 and an+1= 2an + 5, n = 1, 2 .. , then a100 is equal to ?

Question

If  a_1 = 1  and  a_{n+1}= 2a_n + 5 , then  a_{100}  is equal to ?

A.  5*2^{99}-6

B.  5*2^{99}+6

C.  6*2^{99}+5

D.  6*2^{99}-5

E.  6*2^{99}-6

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nick1816 · Accepted Answer

a_{n+1}= 2a_n + 5

a_{(1+1)}= 2a_1 + 5

a_{2}= 2*1 + 5

a_{2}= 7

a_{(1+1)}= 6*2^{1}-5 .......(1)

a_{(2+1)}= 2a_2 + 5

a_{3}= 2*7 + 5

a_{3}= 19

a_{(2+1)}= 6*2^{2}-5 ........(2)
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. and so on
We can generalize  a_{n+1} = 6*2^n -5

a_{100}= 6*2^{99}-5

D

sambitspm · Answer

See the attachment

Kinshook · Answer

If  a_1 = 1  and  a_{n+1}= 2a_n + 5 , then  a_{100}  is equal to ?

Series = {1, 7, 19, 43, 91,....}

In general,

Let us assume

a_n = A*2^{n-1} + B

a_1 = A + B = 1
a_2 = 2A + B = 7
A = 6; B = - 5

a_n = 6*2^{n-1} -5

a_{100} = 6*2^{99} - 5

IMO D

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