A well-known method for approximating √c for a positive real number c consists of the following recurrence relation (based on Newton’s method; see Section 4.8). Let a₀ = c and

a. Use this recurrence relation to approximate √10. How many terms of the sequence are needed to approximate √10 with an error less than 0.01? How many terms of the sequence are needed to approximate √10 with an error less than 0.0001? To compute the error, assume a calculator gives the exact value.

b. Use this recurrence relation to approximate √c, for c = 2, 3, . . . , 10. Make a table showing the number of terms of the sequence needed to approximate √c with an error less than 0.01.

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for n = 0, 1, 2, 3, . . . . An An+1