Question

A certain 300-term geometric sequence has first term 1337 and common ratio $-\frac12$. How many terms of this sequence are greater than 1

Answer 1

There are 6 terms in the sequence greater than 1.

What is a geometric series ?

A geometric series is  sum of infinite numbers which has a common ratio between its successive terms.

The missing common ratio value is -1/2

It is given in the question that

the number of terms in the sequence = 300

Sum = a₁ (1-rⁿ)/(1-r)

nth term is given by

aₙ = a₁r⁽ⁿ⁻ ¹⁾

1337(1/2)^(n - 1) = 1

(1/2)^(n - 1)  = 1/1337

Applying log on both sides

log (1/2)^(n - 1) = log (1/1337)

(n - 1) log(1/2)  = log(1/1337)

n  = log (1/1337)/ log (1/2) + 1 ≈ 11.384

the 11th term  is 1337(-1/2)^(10)  ≈ 1.305

and

And the 12th term is 1337(-1/2)^11  =  -.653

As the even terms are negative , they are less than 1 and , therefore the odd terms from 1 - 11 term will be positive and greater than 1.

Therefore there are 6 terms in the sequence greater than 1.

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