To fill: Check whether the given statement is true or false.
Solution:
It is a true statement.
Given:
It is a geometric sequence with first term a 1 and common ratio, where r ≠ 0, 1 , te sum of the first n terms is S n = a 1 ⋅ 1 − r n 1 − r .
Calculation:
Let { a n } be a geometric sequence with first term a 1 and common ratio r , where r ≠ 0, r ≠ 1 . The sum S n of the first n terms of { a n } = { a 1 r n − 1 } is S n = a 1 + a 1 r + ⋅⋅⋅ + a 1 r n − 1 . -----(1)
Multiply each side by r obtain, r S n = a 1 r + a 1 r 2 + ⋅⋅⋅ + a 1 r n . -----(1)
Subtract (2) from (1). The result becomes S n − r S n = a 1 − a 1 r n .
S n ( 1 − r ) = a 1 ( 1 − r n )
Since r ≠ 1, S n = a 1 ( 1 − r n ) ( 1 − r ) .
Hence the given statement is true.
Precalculus
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Glencoe Math Accelerated, Student Edition
University Calculus: Early Transcendentals (3rd Edition)
Calculus and Its Applications (11th Edition)
Calculus & Its Applications (14th Edition)
Calculus: Early Transcendentals (2nd Edition)