Question

6. The first and the last terms 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

Let a and d are first term and common

Difference for an AP.

Number of terms of AP = n

Last term = nth term = l = an

a = 17

d = 9

l = 350

a + ( n - 1 ) d = 350

17 + ( n - 1 ) 9 = 350

( n - 1 ) 9 = 350 - 17

( n - 1 ) 9 = 333

n - 1 = 333 ÷ 9

n - 1 = 37

n = 37 + 1

n = 38

Therefore ,

Number of terms in given AP = n = 38

Sum of n terms of AP = Sn

Sn = n ÷ 2 ( a + l )

Here n= 38

S38 = 38 ÷ 2 [ 17 + 350 ]

= 19 × 367

= 6973

$Amannnscharlie$

Answer 2

$\large\underline\pink{Given:-}$

• First term of Ap = 17
• Last term of Ap = 350
• Common difference = 9

$\large\underline\pink{To find:-}$

• Sum of all term of Ap ....?

$\large\underline\pink{Solutions:-}$

$\: \: \: \: \: \therefore \: \: {a_n} \: \: = \: \: {a} \: + \: {({n} \: - \: {1})} \: d$

• $\: \: \: \: \: {a} \: \: = \: \: {17}$
• $\: \: \: \: \: {d} \: \: = \: \: {9}$
• $\: \: \: \: \: {n} \: \: = \: \: {?}$
• $\: \: \: \: \: {a_n} \: \: = \: \: {350}$

$\: \: \: \: \: We \: \: have$

$\: \: \: \: \: \therefore \: \: {a_n} \: \: = \: \: {a} \: + \: {({n} \: - \: {1})} \: d$

$\: \: \: \: \: \leadsto \: \: {350} \: \: = \: \: {17} \: + \: {({n} \: - \: {1})} \: {9}$

$\: \: \: \: \: \leadsto \: \: {350} \: \: = \: \: {17} \: + \: {({9n} \: - \: {9})}$

$\: \: \: \: \: \leadsto \: \: {350} \: - \: {17} \: \: = \: \: {9n} \: - \: {9}$

$\: \: \: \: \: \leadsto \: \: {333} \: \: = \: \: {9n} \: - \: {9}$

$\: \: \: \: \: \leadsto \: \: {333} \: + \: {9} \: \: = \: \: {9n}$

$\: \: \: \: \: \leadsto \: \: {347} \: \: = \: \: {9n}$

$\: \: \: \: \: \leadsto \: \: {n} \: \: = \: \: {347}{9}$

$\: \: \: \: \: \leadsto \: \: {n} \: \: = \: \: {38}$

$\: \: \: \: \: \: \: Now, \\ \: \:\therefore \: \: Sum \: \: of \: \: n \: \: term \: \: of \: \: Ap.$

$\: \: \: \: \: \: \: \therefore \: \: {S_n} \: \: \frac{n}{2} \: {({a} \: + \: {a_n})}$

$\: \: \: \: \: \: \: \leadsto \: \: \frac{38}{2} \: {({17} \: + \: {350})}$

$\: \: \: \: \: \: \: \leadsto \: \: {19} \: \times \: {367}$

$\: \: \: \: \: \: \: \leadsto \: \: {6973}$

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

$\: \: \: \: \: \: \: Hence, \\ \: \:\therefore \: \: Sum \: \: of \: \: {38th} \: \: term \: \: of \: \: Ap \: \: is \: \: {6973}$

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