Question

chapter: arithmetic and geometric progression
The first and the last term of an AP are 17 and 350 respectively. If the common
difference is 9, how many terms are there and what is their sum?

Answer 1

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Answer 2

Given :

• First term, a = 17
• Last term, l = 350
• Common difference, d = 9

To Find :

• Number of terms in AP, n = ?
• Sum of total number of terms in AP, $\sf S_{n} = ?$

Solution :

Let, l be the nth term of AP.

$\sf : \implies a_{n} = l = 350$

Now, we know that :

$\Large \underline{\boxed{\bf{ a_{n} = a + ( n - 1 ) d }}}$

By, putting values,

$\sf : \implies 350 = 17 + ( n - 1 ) \times 9$

$\sf : \implies 350 = 17 + 9n - 9$

$\sf : \implies 350 = 8 + 9n$

$\sf : \implies 350 - 8 = 9n$

$\sf : \implies 342 = 9n$

$\sf : \implies \dfrac{ \cancel{342}^{38}}{\cancel{9}} = n$

$\sf : \implies 38 = n$

$\sf : \implies n = 38$

$\large \underline{\boxed{\sf n = 38}}$

Hence, There are 38 number of terms in given AP.

Now, let's find sum of total number of terms in AP.

We know that :

$\Large \underline{\boxed{\bf{ S_{n} = \dfrac{n}{2} ( a + a_{n} ) }}}$

We have :

• n = 38
• a = 17
• $\sf a_{n} = 350$

$\sf : \implies S_{38} = \dfrac{\cancel{38}^{19}}{\cancel{2}} ( 17 + 350 )$

$\sf : \implies S_{38} = 19 (367)$

$\sf : \implies S_{38} = 19 \times 367$

$\sf : \implies S_{38} = 6973$

$\large \underline{\boxed{\sf S_{38} = 6973}}$

Hence, There are 38 number of terms in given AP and their sum is 6973.