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已知首项为 1 的数列

\left\{a_{n}\right\}

满足

a_{n+1}-2 a_{n}=\frac{1-(-1)^{n+1}}{2}

, ...

Mar 8, 2024

已知首项为 1 的数列

\left\{a_{n}\right\}

满足

a_{n+1}-2 a_{n}=\frac{1-(-1)^{n+1}}{2}

, 则数列

\left\{3 a_{2 n}+2\right\}

的前

n

项和

S_{n}=

Solution by Steps

step 1

Find the general term

a_n

by solving the recursive formula

step 2

The recursive formula is

a_{n+1} - 2a_n = \frac{1 - (-1)^{n+1}}{2}

step 3

Notice that

\frac{1 - (-1)^{n+1}}{2}

alternates between 0 and 1 as

n

changes

step 4

For even

n

, the formula simplifies to

a_{n+1} - 2a_n = 0

step 5

For odd

n

, the formula simplifies to

a_{n+1} - 2a_n = 1

step 6

Solve the homogeneous equation

a_{n+1} - 2a_n = 0

to find the general solution for even

n

step 7

The characteristic equation is

r - 2 = 0

, which has a solution

r = 2

step 8

The general solution for even

n

a_n = C \cdot 2^n

step 9

Use the initial condition

a_1 = 1

to find the constant

C

step 10

Since

a_1 = C \cdot 2^1

, we have

C = \frac{1}{2}

step 11

The general term for even

n

a_n = \frac{1}{2} \cdot 2^n

step 12

Now, find the general term for odd

n

using the particular solution

step 13

For odd

n

, we have

a_{n+1} = 2a_n + 1

step 14

Substitute

n

with

2m

to find

a_{2m+1}

step 15

We get

a_{2m+1} = 2a_{2m} + 1 = 2 \cdot \frac{1}{2} \cdot 2^{2m} + 1 = 2^{2m} + 1

step 16

The general term for odd

n

a_n = 2^{n-1} + 1

step 17

Now, find the expression for

3a_{2n} + 2

step 18

Substitute

a_{2n}

with the general term for even

n

step 19

We get

3a_{2n} + 2 = 3 \cdot \frac{1}{2} \cdot 2^{2n} + 2 = 3 \cdot 2^{2n-1} + 2

step 20

Find the sum

S_n

of the sequence

3a_{2n} + 2

from

n=1

n

step 21

The sum of a geometric series is

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}

, where

r

is the common ratio

step 22

The first term

a_1

3 \cdot 2^{2 \cdot 1 - 1} + 2 = 3 \cdot 2 + 2 = 8

step 23

The common ratio

r

2^{2} = 4

step 24

The sum is

S_n = 8 \cdot \frac{1 - 4^n}{1 - 4}

step 25

Simplify the sum to get

S_n = 8 \cdot \frac{4^n - 1}{3}

Answer

S_n = \frac{8}{3} \cdot (4^n - 1)

Key Concept

Recursive sequences and geometric series sum

Explanation

The problem involves finding a general term for a recursive sequence and then using that term to determine the sum of a related sequence. The sum of the new sequence, which is geometric, is found using the formula for the sum of a geometric series.