Let a_ 1 , a_ 2 , , a_ 10 be an AP with common difference -3 and … - Vidyadip

Question

Let $a_{1}, a_{2}, \ldots, a_{10}$ be an $AP$ with common difference $-3$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{10}$ be a $GP$ with common ratio $2.$ Let $c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10 .$ If $c_{2}=12$ and $\mathrm{c}_{3}=13$, then $\sum_{\mathrm{k}=1}^{10} \mathrm{c}_{\mathrm{k}}$ is equal to ..... .

✓

Answer

a
$c_{2}=a_{2}+b_{2}=a_{1}-3+2 b_{1}=12$

$a_{1}+2 b_{1}=15....(1)$

$c_{3}=a_{3}+b_{3}=a_{1}-6+4 b_{1}=13$

$a_{1}+4 b_{1}=19....(2)$

from $(1)\, \,(2) b_{1}=2, a_{1}=11$

$\sum_{k=1}^{10} c_{k}=\sum_{k=1}^{10}\left(a_{k}+b_{k}\right)=\sum_{k=1}^{10} a_{k}+\sum_{k=1}^{10} b_{k}$

$=\frac{10}{2}(2 \times 11+9 \times(-3))+\frac{2\left(2^{10}-1\right)}{2-1}$

$=5(22-27)+2(1023)$

$=2046-25=2021$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\operatorname{gcd}( m , n )=1$ and $1^2-2^2+3^2-4^2+\ldots \ldots$ $+(2021)^2-(2022)^2+(2023)^2=1012 m ^2 n$, then $m ^2- n ^2$ is equal to

If $a, b, c$ be positive real numbers and the value of $\theta = {\tan ^{ - 1}}\sqrt {\frac{{a(a + b + c)}}{{bc}}} + {\tan ^{ - 1}}\sqrt {\frac{{b(a + b + c)}}{{ca}}} + {\tan ^{ - 1}}\sqrt {\frac{{c(a + b + c)}}{{ab}}} $,then $\tan \theta $ is equal to

The sum of the square of the modulus of the elements in the set

$\{z=a+i b: a, b \in Z, z \in C,|z-1| \leq 1,|z-5| \leq|z-5 i|\}$ is ........

Let the locus of the points from which the tangents drawn to $y = x^2$ make an angle of $45^o$ with each other is $16y^2 - 16x^2 + ky + 1 = 0$ then $k$ is equal to

Let $a >0$ be a root of the equation $2 x ^2+ x -2=0$. If $\lim _{x \rightarrow \frac{1}{a}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{\left(1-a x^2\right)}=\alpha+\beta \sqrt{17}$, where $\alpha, \beta \in Z$ then $\alpha+\beta$ is equal to $.........$

Let the volume of tetrahedron $ABCD$ is $81$ cubic units $\&$ $G_1,G_2,G_3$ are centroids of triangular faces $ABC, ABD \, \& \,ACD$ respectively, then volume of tetrahedron $A\,G_1G_2G_3,$ is (in cubic units)

The sum of the series

$\frac{1}{{1 + \sqrt 2 }} + \frac{1}{{\sqrt 2 + \sqrt 3 }} + \frac{1}{{\sqrt 3 + \sqrt 4 }} + .....$ upto $15$ terms is

Let $P_1$ : $y = -x^2 + 4x + 2$ and $P_2$ : ${\rm{x^2 + 5x + }}\frac{{17}}{8} = y$ are two parabolas, then number of common tangents of $P_1$ and $P_2$ is

If one end of a focal chord of the parabola, $y^2 = 16x$ is at $(1, 4),$ then the length of this focal chord is

If $f:R \to \left[ {0,\infty } \right)$ be such that $\mathop {{\rm{lim}}}\limits_{x \to 5} \;f\left( x \right)$ exists and $\mathop {{\rm{lim}}}\limits_{x \to 5} \frac{{{{\left( {f\left( x \right)} \right)}^2} - 9}}{{\sqrt {\left| {x - 5} \right|} }} = 0\;$,then $\mathop {{\rm{lim}}}\limits_{x \to 5} f\left( x \right)$ equals :