how-do-we-find-the-sum-of-the-terms-after-the-n-th-term-of-a-GP-

Question Number 163675 by alcohol last updated on 09/Jan/22

h o w d o w e f i n d t h e s u m o f t h e t e r m s a f t e r

t h e n^{t h} t e r m o f a G P

Commented by mr W last updated on 09/Jan/22

m a k e t h e {(n + 1)}^{t h} t e r m t o t h e f i r s t

t e r m a n d a p p l y t h e f o r m u l a f o r s u m

f r o m f i r s t t e r m t o \infty .

a_{1} + a_{1} q + a_{1} q^{2} + \dots + a_{1} q^{n - 1} + a_{1} q^{n} + a_{1} q^{n + 1} + \dots = \frac{a_{1}}{1 - q}

a_{1} q^{n} + a_{1} q^{n + 1} + \dots = \frac{a_{1} q^{n}}{1 - q}

o r

s u m f r o m f i r s t t o n^{t h} t e r m : \frac{a_{1} (1 - q^{n})}{1 - q}

s u m f r o m f i r s t t e r m t o \infty : \frac{a_{1}}{1 - q}

s u m f r o m {(n + 1)}^{t b} t e r m t o \infty : \frac{a_{1}}{1 - q} - \frac{a_{1} (1 - q^{n})}{1 - q} = \frac{a_{1} q^{n}}{1 - q}

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