JEE MAIN - Mathematics (2021 - 31st August Morning Shift - No. 19)
The mean of 10 numbers 7 $$\times$$ 8, 10 $$\times$$ 10, 13 $$\times$$ 12, 16 $$\times$$ 14, ....... is ____________.
Answer
398
Explanation
7 $$\times$$ 8, 10 $$\times$$ 10, 13 $$\times$$ 12, 16 $$\times$$ 14 ........
Tn = (3n + 4) (2n + 6) = 2(3n + 4) (n + 3)
= 2(3n2 + 13n + 12) = 6n2 + 26n + 24
S10 = $$\sum\limits_{n = 1}^{10} {{T_n}} = 6\sum\limits_{n = 1}^{10} {{n^2}} + 26\sum\limits_{n = 1}^{10} n + 24\sum\limits_{n = 1}^{10} 1 $$
$$ = {{6(10 \times 11 \times 21)} \over 6} + 26 \times {{10 \times 11} \over 2} + 24 \times 10$$
$$ = 10 \times 11(21 + 13) + 240$$
= 3980
Mean $$ = {{{S_{10}}} \over {10}} = {{3980} \over {10}} = 398$$
Tn = (3n + 4) (2n + 6) = 2(3n + 4) (n + 3)
= 2(3n2 + 13n + 12) = 6n2 + 26n + 24
S10 = $$\sum\limits_{n = 1}^{10} {{T_n}} = 6\sum\limits_{n = 1}^{10} {{n^2}} + 26\sum\limits_{n = 1}^{10} n + 24\sum\limits_{n = 1}^{10} 1 $$
$$ = {{6(10 \times 11 \times 21)} \over 6} + 26 \times {{10 \times 11} \over 2} + 24 \times 10$$
$$ = 10 \times 11(21 + 13) + 240$$
= 3980
Mean $$ = {{{S_{10}}} \over {10}} = {{3980} \over {10}} = 398$$
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