Question

Find the sums of 7+10+13+........184

Answer 1

Given ,

• First term (a) = 7
• Common difference (d) = 3
• Last term (l) = 184

We know that , the first n terms of an AP is given by

$\star \: \sf \: a_{n} = a + (n - 1)d$

Thus ,

$\sf \Rightarrow </p><p></p><p>184 = 7 + (n - 1)3 \\ \\ \sf \Rightarrow \frac{177}{3} = (n - 1) \\ \\ \sf \Rightarrow </p><p> n - 1 = 59 \\ \\ \sf \Rightarrow </p><p>n = 60$

And the sum of first n terms of an AP is given by

$\star \: \: \sf S_{n} = \frac{n}{2} \times (a + l)$

Thus ,

$\sf \Rightarrow </p><p></p><p>S = \frac{60}{2} \times (7 + 184) \\ \\ \sf \Rightarrow </p><p></p><p>S = 30 × (191) \\ \\ \sf \Rightarrow </p><p> </p><p>S = 5730</p><p>$

$\therefore \underline{ \bold{ The \: sum \: of \: first \: 60 \: terms \: of \: AP \: is \: 5730}}$

Answer 2

Question :–

• Find the sum of 7 + 10 + 13 + ..... + 184

ANSWER :–

$\\ \longrightarrow { \red { \boxed{ \bold{ S_{60} = 5,730 }}}}$

GIVEN :–

• An A.P. 7 + 10 + 13 + ........ + 184

TO FIND :–

• Sum of given series = ?

SOLUTION :–

• We know that Sum of A.P. is –

$\\ \implies { \pink{ \boxed{ \boxed{ \bold{ S_{n} = \dfrac{n}{2}[2a + (n - 1)d] }}}}}$

• nth term of A.P. –

$\\ \implies { \pink{ \boxed{ \boxed{ \bold{ T_{n} = [a + (n - 1)d] }}}}}$

• Here –

$\\ { \blue{ \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: a = first \: \: term }}}$

$\\ { \blue{ \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: d = common \: \: difference }}}$

$\\ { \blue{ \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: n = total \: \: term }}}$

$\\ { \blue{ \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: S_{n} = sum \: \: of \: \: n \: \: \: terms }}}$

$\\ { \blue{ \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: T_{n} = nth \: \: term }}}$

• In the given series –

$\\ { \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: a = 7 }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: d = 3 }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: \: \: \: . \: \: T_{n} = 184 }}$

• Now put the values –

$\\ \implies { \bold{ 184 = [7 + (n - 1)3] }}$

$\\ \implies { \bold{ 184 = 7 +3n - 3 }}$

$\\ \implies { \bold{ 184 = 4 +3n}}$

$\\ \implies { \bold{3n = 180}}$

$\\ \implies { \boxed{ \bold{n = 60}}}$

• Now sum –

$\\ \implies { \bold{ S_{60} = \dfrac{60}{2}[2(7) + (60 - 1)(3)] }}$

$\\ \implies { \bold{ S_{60} = 30[14 + 177] }}$

$\\ \implies { \bold{ S_{60} = 30[191] }}$

$\\ \implies { \boxed{ \bold{ S_{60} =5,730 }}}$