Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its term is : [2025]

Question

Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{th}$ term is : [2025]

Smart Achievers · Accepted Answer

(2)

We have, $a_{1} + a_{2} + a_{3} = 54$

$\Rightarrow 3 (a + d) = 54 \Rightarrow a + d = 18$

$S_{20} = 10 [2 a + 19 d] = 10 [36 + 17 d]$

As $1600 < S_{20} < 1800$

$\Rightarrow 1600 < 10 (36 + 17 d) < 1800$

$\Rightarrow 160 < 36 + 17 d < 180$

$\Rightarrow 124 < 17 d < 144$

$\Rightarrow 7 \frac{5}{17} < d < 8 \frac{8}{17}$

Since, A.P. is of positive integers.

Common difference will be a natural number

$∴ d = 8 \Rightarrow a = 10$

$∴ a_{11} = 10 + 10 \times 8 = 90$ .

Solution

Q.
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{th}$ term is : [2025]

1 84

2 90

3 108

4 122

Want Us to Email you About Special Offers & Updates?