Question

IF 3+5+7+9+....... + upto n terms =288 then find n .​

Answer 1

$\underline{\underline{\huge{\gray{\tt{\textbf Answer :-}}}}}$

$\:\:$

$\sf\large\bold{\orange{\underline{\blue{ Given :-}}}}$

$\:\:$

• 3 + 5 + 7 + 9+....... + upto n terms = 288

$\:\:$

$\sf\large\bold{\orange{\underline{\blue{ To \: Find :-}}}}$

$\:\:$

• Value of n

$\:\:$

$\sf\large\bold{\orange{\underline{\blue{ Solution :-}}}}$

$\:\:$

Clearly they could be placed in an AP as they have equal difference between them

$\:\:$

➠ 3 , 5 , 7 , 9 , --------- nth term

$\:\:$

We know that ,

$\:\:$

$\underline{\bold{\texttt{Sum of n terms :}}}$

$\:\:$

➠ $\sf S_n = \dfrac { n } { 2 } (2a + (n - 1)d)$ ⚊⚊⚊⚊ ⓵

$\:\:$

Where ,

$\:\:$

• $\sf S_n$ = nth term

• n = Number of terms

• a = First term

• d = Common difference

$\:\:$

$\underline{\bold{\texttt{For given AP :}}}$

$\:\:$

• $\sf S_n$ = 288

• n = n

• a = 3

• d = 5 - 3 = 2

$\:\:$

⟮ Putting the above values in ⓵ ⟯

$\:\:$

➜ $\sf S_n = \dfrac { n } { 2 } (2a + (n - 1)d)$

$\:\:$

➜ $\sf 288 = \dfrac { n} { 2 } (2(3) + (n - 1)2)$

$\:\:$

➜ $\sf 288 = \dfrac { n} { 2 } (6 + 2n - 2)$

$\:\:$

➜ 288 × 2 = n(2n + 4)

$\:\:$

➜ 576 = 2n² + 4n

$\:\:$

⟮ Dividing the above equation by 2 ⟯

$\:\:$

➜ 288 = n² + 2n

$\:\:$

➜ n² + 2n - 288 = 0

$\:\:$

➜ n² + 18n -16n - 288 = 0

$\:\:$

➜ n(n + 18) -16(n + 18) = 0

$\:\:$

➜ (n + 18)(n - 16) = 0

$\:\:$

• n = -18
• n = 16

$\:\:$

As term can't be negative hence ,

$\:\:$

➨ n = 16

$\:\:$

∴ The value of n is 16

$\:\:$