Consider the Recursive Sequence Defined By $a _ { 1 } = 1 ; a _ { n + 1 } = \frac { 1 } { 2 } \left( a _ { n } + 4 \right) , n > 1$

question 154

Essay

Consider the recursive sequence defined by $a _ { 1 } = 1 ; a _ { n + 1 } = \frac { 1 } { 2 } \left( a _ { n } + 4 \right) , n > 1$ .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

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Definitions:

Accrual Accounting

An accounting method where transactions are recorded when they are earned or incurred, not necessarily when cash changes hands, providing a more accurate financial picture.

Expense Recognition

A principle in accounting that dictates when costs are recognized as expenses in the financial statements, often aligned with the revenue they helped to generate.

Cash Basis

An accounting method where revenues and expenses are recorded when cash is received or paid, not when the obligation is incurred.

Comparability

An accounting principle that ensures financial statements can be compared between periods and across companies.