Answer

Correct Answer : b ) 3¹⁰ – 6

Let S = 7 x 3 + 9 x 9 + 11 x 27 + 13 x 81 + ……….. + up to 7 term

Or, S = 7 x 3 + 9 x 3² + 11 x 3³ + 13 x 3⁴ + ……….. + 19 x 3⁷ ……….. (1)

Now, Multiplying the above equation (1) by ‘3’ on both sides

3S = 7 x 3² + 9 x 3³ + 11 x 3⁴ + ……….. + 19 x 3⁸ ………… (2)

Now subtracting equation (1) by equation (2) we get

-2S = 21 + 2 x 3² + 2 x 3³ + ………. + 2 x 3⁷ – 19 x 3⁸

-2S = 21 + 2 (3² + 3³ + ………. + 3⁷)  – 19 x 3⁸

We know that sum of a G.P. having common ratio (r > 1) will be given as

\(Sum=\frac{a(r^{n}-1)}{(r-1) }\)

Where a is the first term of the geometric progression (G.P.)
r is the common difference
and, n is the number of terms
Therefore,
-2S= 21 + 2(3² (3ⁿ-1)/(3-1)) – 19*3⁸

-2S = 21 + 2(9(3⁶-1)/(3-1)) – 19*3⁸

-2S = 21 + 9(3⁶-1)-19*3⁸

-2S = 21 + 9*(3⁶ )-9-19*3⁸

-2S = 21 - 9 + 9*(3⁶ ) - 19*3⁸

-2S = 12 + (3⁸ ) - 19*3⁸

-2S = 12 + (1-19)*3⁸

-2S = 12 - 18*3⁸

S= 9*3⁸ - 6

S= 3¹⁰ - 6

Hence, (b) is the correct answer.
Such type of question is asked in various government exams like SSC CGL, SSC MTS, SSC CPO,  SSC CHSL, RRB JE, RRB NTPC, RRB GROUP D, RRB OFFICER SCALE-I, IBPS PO, IBPS SO, RRB Office Assistant, IBPS Clerk, RBI Assistant, IBPS RRB OFFICER SCALE 2&3, etc. 
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