Question

The first term of a sequence is $x_{1}=1 .$ Each succeeding term is the sum of all those that come before it: $$ x_{n+1}=x_{1}+x_{2}+\cdots+x_{n} $$ Write out enough early terms of the sequence to deduce a general formula for $x_{n}$ that holds for $n \geq 2$

          The first term of a sequence is $x_{1}=1 .$ Each succeeding term is
the sum of all those that come before it:
$$
x_{n+1}=x_{1}+x_{2}+\cdots+x_{n}
$$
Write out enough early terms of the sequence to deduce a general
formula for $x_{n}$ that holds for $n \geq 2$

Added by Francisco Javier J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/04/2024

Step 1

Now let's find the next few terms: $x_2 = x_1 = 1$ $x_3 = x_1 + x_2 = 1 + 1 = 2$ $x_4 = x_1 + x_2 + x_3 = 1 + 1 + 2 = 4$ $x_5 = x_1 + x_2 + x_3 + x_4 = 1 + 1 + 2 + 4 = 8$ We can see a pattern emerging: $x_n = 2^{n-2}$ for $n \geq 2$. Let's prove this by Show more…

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The first term of a sequence is $x_{1}=1 .$ Each succeeding term is the sum of all those that come before it: $$ x_{n+1}=x_{1}+x_{2}+\cdots+x_{n} $$ Write out enough early terms of the sequence to deduce a general formula for $x_{n}$ that holds for $n \geq 2$