Download App

Arithmetic Progressions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsArithmetic Progressions3 Marks
Question
If $\text{s}_\text{n}=\text{n}^2\ \text{p}$ and $\text{s}_\text{m}=\text{m}^2\ \text{p},\ \text{m}\not=\text{n},$ in an A.P., prove that $\text{s}_\text{p}=\text{p}^3.$
✓

Answer

Let a be the first term of the AP and d is the common difference. then
$\text{s}_\text{n}=\frac{\text{n}}{2}(2\text{a}+(\text{n}-1)\text{d})$
$\text{n}^2\text{p}=\frac{\text{n}}{2}(2\text{a}+(\text{n}-1)\text{d})$
$\text{np}=\frac{1}{2}[2\text{a}+(\text{n}-1)\text{d}]$
$2\text{np}=2\text{a}+(\text{n}-1)\text{d}\ .....{(1)}$
Again
$\text{s}_\text{m}=\frac{\text{m}}{2}(2\text{a}+(\text{m}-1)\text{d})$
$\text{m}^2\text{p}=\frac{\text{m}}{2}(2\text{a}+(\text{m}-1)\text{d})$
$\text{mp}=\frac{1}{2}[2\text{a}+(\text{m}-1)\text{d}]\ .....{(2)}$
Now subtract (1) from (2)
$2\text{p}(\text{m}-\text{n})=(\text{m}-\text{n})\text{d}$
$\text{d}=2\text{p}$
Therefore
$2\text{mp}=2\text{a}+(\text{m}-1)\times2\text{p}$
$2\text{a}=2\text{p}$
$\text{a}=\text{p}$
The sum up p terms will be:
$\text{s}_\text{p}=\frac{\text{p}}{2}(2\text{a}+(\text{p}-1)\text{d})$
$=\frac{\text{p}}{2}(2\text{p}+(\text{p}-1).2\text{p})$
$=\frac{\text{p}}{2}(2\text{p}+2\text{p}^2-2\text{p})$
$=\text{p}^3$

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 Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Let $A = \{3, 6, 12, 15, 18, 21\}, B = \{4, 8, 12, 16, 20\}, C = \{2, 4, 6, 8, 10, 12, 14, 16\}, D = \{5 , 10, 15, 20\}.$ Find:
  1. $A - B$
  2. $A - C$
  3. $A - D$
  4. $B - A$
  5. $C - A$
  6. $D - A$
  7. $B - C$
  8. $B - D.$
→
If $\cot \text{x}(1+\sin\text{x})=4\text{ m}$ and $\cot \text{x}(1-\sin\text{x})=4\text{ n},$prove that $\text{(m}^2-\text{n}^2)^2=\text{mn.}$
→
The angles of a triangle are in A.P. and the number of degrees in the least angle is to the number of degrees in the mean angle as 1 : 120. Find the angles in radians.
→
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If there are 640 litres of the 8% solution, how many litres of 2% solution will have to be added?
→
The angle in one regular polygon is to that in another as 3 : 2 and the number of sides in first is twice that in the second. Determine the number of sides of two polygons.
→
Show the following quadratic equation by factorization method:
$\text{x}^2 +2\text{ x} +\frac{3}{2}=0$
→
Find the coefficient of:
x in the expansion of $(1+3\text{x}+7\text{x}^2)(1-\text{x})^{16}.$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos7\text{x}}{\text{x}^2}$
→
Differentiate the following from the first principle$\text{e}^{\sqrt{2\text{x}}}$
→
If $S_p$ denotes the sum of the series $1+\text{r}^{\text{p}}+\text{r}^{2\text{r}}+\ \dots\text{ to }\infty$ and $S_p$ the sum of the series $1-\text{r}^{\text{p}}+\text{r}^{\text{2p}}-\ \dots\text{ to }\infty,$ prove that $\text{S}_\text{p} + \text{S}_\text{p} = 2 \text{S}_{2\text{p}}.$
→