Question

2. In an arithmetic sequence, first term is 5
and square of seventh term is 529. Find the
common difference of the sequence. Find
sum of first 7 terms.​

Answer 1

Answer:

d=3

7th term= -63

Hope this helps u

Answer 2

GIVEN :-

• First term , $\sf a_1 = 5$

• Square of 7th term , $\sf (a_7)^2 = 529$

TO FIND :-

• Common difference of the AP .

• Sum of first 7 terms .

SOLUTION :-

  Square of seventh term = 529

        $\sf a_7 = a+16d$

   $\hookrightarrow \sf (a_7)^2 = 529 \\\\\hookrightarrow (a+6d)^2 = 529 \\\\\hookrightarrow a^2+ 36d^2+12ad = 529 \\\\\hookrightarrow 5^2+36d^2+12\times5\times d=529\\\\\hookrightarrow 25+36d^2+60d = 529 \\\\\hookrightarrow 36d^2+60d = 529-25\\\\\hookrightarrow 36d^2+60d = 504 \\\\\hookrightarrow 12(3d^2+5d) = 504 \\\\\hookrightarrow 3d^2+5d = 42 \\\\\hookrightarrow 3d^2+5d-42 = 0\\\\\hookrightarrow 3d^2-9d+14d-42 = 0 \\\\\hookrightarrow 3d(d^2-3)+14(d-3)=0\\\\\hookrightarrow (3d+14)(d-3)=0$

   $\bf d= 3 \ , \ d = -\dfrac{14}{3}$

  Sum of first n terms =  $\sf \dfrac{n}{2}\times (2a+(n-1)d)$

• Sum of first 7 terms of  AP having common difference 3 .

           

          $\hookrightarrow \dfrac{7}{2}\times (2 \times 5+ (7-1)\times 3)\\\\\hookrightarrow \dfrac{7}{2}\times (10+18)\\\\\hookrightarrow \dfrac{7}{2}\times28 \\\\\hookrightarrow \bf 98$

• Sum of first 7 terms of  AP having common difference $\bf- \dfrac{14}{3}$ .

 

        $\hookrightarrow \sf \dfrac{7}{2}\times (2\times 5+(7-1)\times - \dfrac{14}{3})$

       $\hookrightarrow \sf \dfrac{7}{2}\times (10 -28)$

       $\hookrightarrow \sf \dfrac{7}{2} \times -18$

       $\hookrightarrow \bf -63$