Answer :

We can write this as (2 + 3^{1})+(2+3^{2}) + (2 +3^{3})+… to 10 terms

= ( 2+2+2+… to 10 terms) + ( 3+3^{2}+3^{3}+… to 10 terms)

= 2×10 + (3+3^{2}+3^{3}+… to 10 terms)

= 20 + (3+3^{2}+3^{3}+… to 10 terms)

Sum of a G.P. series is represented by the formula, , when r≠1. ‘S_{n}’ represents the sum of the G.P. series upto n^{th} terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here,

a = 3

r =(ratio between the n term and n-1 term) 3

n = 10 terms

⇒

⇒

⇒

Thus, sum of the given expression is

= 20 + (3+3^{2}+3^{3}+… to 10 terms)

= 20 + 88572

=88592

(ii) The given expression can be written as,

( 2^{1} + 3^{1-1}) + (2^{2} + 3^{2-1}) + …to n terms

= (2 + 3^{0}) + ( 2^{2}+ 3^{1}) + …to n terms

= (2 + 1) +(2^{2} + 3 ) + …to n terms

= (2 + 2^{2} + …to terms) + ( 1 + 3 + … to terms)

Sum of a G.P. series is represented by the formula, , when r≠1. ‘S_{n}’ represents the sum of the G.P. series upto n^{th} terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here,

a = 2, 1

r = (ratio between the n term and n-1 term)2, 3

terms

⇒

⇒

(iii) We can rewrite the given expression as

( 5^{1} + 5^{2} + 5^{3}+ …to 8 terms)

Sum of a G.P. series is represented by the formula, , when r>1. ‘S_{n}’ represents the sum of the G.P. series upto n^{th} terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here,

a = 5

r =(ratio between the n term and n-1 term) 5

n = 8 terms

⇒

⇒

⇒

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