Question

The 1st and tha last term of an AP are 17 and 350 respectively if tha common different is 9.than how many are terms are there and what is their sum​

Answer 1

Given:-

$\implies \sf 1^{st} term\: of \: Ap = 17 \\ \implies \sf last \: term \: of\: Ap = 350 \\ \implies \sf Common \: difference = 9$

To find:-

Terms and their sum.

Solution:-

$\implies \sf Here, a (First\: term) = 17 \\ \implies \sf a_{n} (Last \: term) = 350 \\ \implies \sf d (Common \: Difference) = 9 \\ \implies \sf n (terms) = ( ? ) \\ \implies \sf S_{n} (Sum \: of \: terms) = ( ? )$

According to the formula for finding the the terms we will know the value of n.

$\large {\boxed {\underline {\sf {\red {a_{n} = a + ( n - 1 ) \times d }}}}}$

Putting the values according to question we will get.

$\implies \sf a_{n} = a + (n - 1) \times d \\ \\ \implies \sf 350 = 17 + (n - 1) \times 9 \\ \\ \implies \sf 350 - 17 = (n - 1) \times 9 \\ \\ \implies \sf 333 = (n - 1) \times 9 \\ \\ \implies \sf \dfrac{333}{9} = (n - 1) \\ \\ \implies \sf n = 37 + 1 \\ \\ \implies n = 38$

So, the $\sf n^{th}$ term of an AP is 38 .

Now, putting this value in following formula we will get the sum of AP.

$\large {\underline {\boxed {\sf {\red {S_{n} = \dfrac{n}{2} [ a + a_{n} ] }}}}}$

Putting the values we will get.

$\implies \sf S_{n} = \dfrac{n}{2} [ a + a_{n} ] \\ \\ \implies \sf S_{n} = \dfrac{38}{2} [ 17 + 350] \\ \\ \implies \sf 19(367) \\ \\ \implies \sf = 6973$

Hence, there are 38 terms in Ap having their sum 6973.

Answer 2

Step-by-step explanation:

Let " a " be the first term of the AP and the last term be $\rm a_{n}$.

" d " be the common difference and " n " be the number of terms

• a = 17, d = 9

⇢ $\rm a_{n} = 350$

⇢ a + (n - 1)d = 350

⇢ 17 + (n - 1) × 9 = 350

⇢ 17 + 9n - 9 = 350

⇢ 9n + 8 = 350

⇢ 9n = 350 - 8

⇢ 9n = 342

⇢ n = 342/9 = 38

∴ Number of terms = 38

Now, sum of 38 terms

$\sf \dashrightarrow S_{n} = \dfrac{n}{2}\bigg[ 2a + (n - 1)d\bigg]$

$\sf \dashrightarrow S_{38} = \dfrac{38}{2}\bigg[ 2\times 17 + (38 - 1)\times 9\bigg]$

$\sf \dashrightarrow S_{38} = 19 \big( 34 + 37\times 9\big)$

$\sf \dashrightarrow S_{38} = 19 \big( 34 + 333\big)$

$\sf \dashrightarrow S_{38} = 19 \times 367$

$\sf \dashrightarrow S_{38} = 6973$

∴ Sum of their terms = 6973