Evaluate: _ k = 1 ^ 11 ( 2 + 3 ^ k ) - Vidyadip

Question

Evaluate:$\sum \limits_ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$

✓

Answer

Given:$\sum _ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$
= (2 + 3¹) + (2 + 3²) + (2 + 3³) + (2 + 3¹¹)
= ( 2 + 2 + 2 +........11 times) + (3 + 3² + 3³ +....... +3¹¹)

= 22 + (3 + 3² + 3³ +....... +3¹¹) ……….(i)

Here 3, 3²,3³ ....... ,3¹¹is in G.P.

$\therefore$a = 3 and r = $\frac { 3 ^ { 2 } } { 3 } = 3$
$\mathrm { S } _ { n } = \frac { 3 \left( 3 ^ { 11 } - 1 \right) } { 3 - 1 } = \frac { 3 } { 2 } \left( 3 ^ { 11 } - 1 \right)$

Putting the value of S_n in eq. (i), we get $\sum _ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right) = 22 + \frac { 3 } { 2 } \left( 3 ^ { 11 } - 1 \right)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Sequences and Series→Sequences and Series — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\sin\text{A}=\frac{12}{13}$ and $\sin\text{B}=\frac{4}{5}$ Where $\frac{\pi}{2}<\text{A}<\pi$and $0<\text{B}<\frac{\pi}{2},$ find the following:

Evaluate the following:

$(2+\sqrt3)^7+(2-\sqrt3)^7$

If $\text{f(x)}=(\text{a}-\text{x}^\text{n})^{\frac{1}{\text{n}}},\text{a}>0$ and $\text{n}\in\text{N},$ then prove that f(f(x)) = x for all x.

Let $f=\left\{\left(x, \frac{x^2}{1+x^2}\right): x \in R\right\}$ be a function from R into R . Determine the range of f .

If sum of the perpendicular distances of a variable point P(x, y) from the lines x + y - 5 = 0 and 3x - 2y + 7 = 0 is always 10. Show that P must move on a line.

The p^th, q^th and r^th terms of an A.P. are a, b, c respectively. Show that (q - r)a + (r - p)b + (p - q)c = 0.

Find the value of $\theta$ and p, if the equation $\text{x} \cos\theta + \text{y} \sin \theta = \text{p}$ is the normal form of the line $\sqrt{3}\text{x}+\text{y}+2=0.$

Prove the following statement by principle of mathematical induction:
For any natural number n, 7ⁿ - 2ⁿ is divisible by 5.

Find the equation of the straight line which makes a triangle of area $96\sqrt{3}$ with the axes and perpendicular from the origin to it makes an angle of 30° with Y-axis.

If each of the observation x₁, x₂,..., x_n is increased by a, where a is a negative or positive number, then show that the variance remains unchanged.