Question

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

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Sum\:of\:7th\:term_{(when\:d=3)}=98}}}$

$\green{\tt{\therefore{Sum\:of\:7th\:term_{(when\:d=\frac{-14}{3})}=-63}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given:}} \\ \tt: \implies First \:term(a_{1} ) = 5 \\ \\ \tt: \implies Square \: of \: 7th \: term(a_{7})^{2} = 529 \\ \\ \red{\underline \bold{To \: Find:}} \\ \tt: \implies Sum \: of \: 7\: term(s_{7}) = ?$

• According to given question :

$\tt: \implies (a_{7})^{2} = 529 \\ \\ \tt: \implies (a + 6d)^{2} = 529 \\ \\ \tt: \implies {a}^{2} + 36 {d}^{2} + 12ad = 529 \\ \\ \tt: \implies {5}^{2} + 36 {d}^{2} + 12 \times 5 \times d = 529 \\ \\ \tt: \implies25 + 36 {d}^{2} + 60d = 529 \\ \\ \tt: \implies36 {d}^{2} + 60d = 529 - 25 \\ \\ \tt: \implies 12( 3{d}^{2} + 5d) = 504 \\ \\ \tt: \implies {3d}^{2} + 5d = 42 \\ \\ \tt: \implies {3d}^{2} + 5d - 42 = 0 \\ \\ \tt: \implies {3d}^{2} - 9d + 14d - 42 = 0 \\ \\ \tt: \implies 3d(d - 3) + 14(d - 3) = 0 \\ \\ \tt: \implies(3d + 14)(d - 3) = 0 \\ \\ \green{\tt: \implies d = 3 \: and \: \frac{ - 14}{3} } \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies s_{n} = \frac{n}{2} (2a + (n - 1)d)$

$\tt\circ \: First \: term = 5 \\ \\ \tt\circ \: Common \: difference = 3 \\ \\ \tt\circ \: Number \: of \: term = 7 \\ \\ \tt: \implies s_{7} = \frac{7}{2} (2 \times 5 + (7 - 1) \times 3) \\ \\ \tt: \implies s_{7} = \frac{7}{2} (10 + 18) \\ \\ \tt: \implies s_{7} = \frac{7}{2} \times 28 \\ \\ \green{\tt: \implies s_{7} = 98} \\ \\ \bold{For \: another \: Common \: difference} \tt\circ \: Common \: difference = \frac{ - 14}{3} \\ \\ \tt: \implies s_{7} = \frac{7}{2} (2 \times 5 + (7 - 1) \times \frac{ - 14}{3} ) \\ \\ \tt: \implies s_{7} = \frac{7}{2} (10 - 28) \\ \\ \green{\tt: \implies s_{7} = - 63}$

Answer 2

$</p><p>\huge{\tt{\pink{Hello!!!}}}$

$</p><p>\tt{\purple{Given \: - }}$

$\tt{ \red{ = > a \: = 5 \: }}$

$</p><p>\tt{\blue{=>{a_{7}}^2 \: = 529 }}$

$</p><p>\tt{\green{Let \: common \: difference \: be \: d. }}$

Hence :

$\tt{ \orange{f(x) = {(6d + 5)}^{2} = 529}}$

Solving the above question we get :

$</p><p>\purple{f(x) => \begin{cases}</p><p></p><p>d = 3 \\</p><p></p><p>d = \frac{-14}{3} </p><p></p><p>\end{cases} }$

$</p><p>\huge{\boxed{\boxed{\red{S = 2a + (n-1)d }}}}$

$</p><p>\tt{\blue{=>n = 7}}$

Placing seperate values of d and solving for each we get :

$</p><p>\purple{Sum \: Till \: 7 \:terms = >\begin{cases}</p><p></p><p>S_{7} = 98 \\</p><p></p><p>S_{7}= -63</p><p></p><p>\end{cases} }$