Question

the first term of an a.p is 3 and the last term is 17.if the sum of all terms is 150,what is the fifth term?​

Answer 1

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Answer 2

Given :-

• first term ( a ) = 3
• last term ( l ) = 17
• sum of all terms ( s_n ) = 150

To find :-

• fifth term of the A.P .

Formula used :-

1. $\boxed{ \sf \: s_n = \frac{n}{2} \: ( a+l ) }$

2.$\boxed{ \sf \: s_n = \dfrac{n}{2} \bigg(2a + (n - 1) \bigg)}$

Solution :-

$\boxed{ \sf \: s_n = \frac{n}{2} \: ( a+l ) }\\ \\ : \implies \sf150 = \frac{n}{2} \: (3 + 17) \\ \\ : \implies \sf \: 300 = 20n \\ \\ : \implies \sf \: n = \frac{300}{20} \\ \\ : \implies \sf \: n = \frac{ \cancel{30 }\cancel0}{ \cancel2 \cancel0} \\ \\ : \implies \underline{\boxed{ \sf \: n =15}}$

Now, we have the value of n ( number of terms ) .

we need to find out the common difference 'd' .

$\boxed{ \sf \: s_n = \dfrac{n}{2} \bigg(2a +\big (n - 1\big) \bigg)} \\ \\ : \implies \sf \: 150 = \frac{15}{2} \bigg(2 \times 3 + \big(15 - 1\big)d\bigg) \\ \\ : \implies \sf \:15 0= \frac{15}{2} \: \big (6 + 14d\big) \\ \\ : \implies \sf \: 300 = 90 +21 0d \\ \\ : \implies \sf \: 300 - 90 = 210d \\ \\ : \implies \sf \: 210 = 210d \\ \\ : \implies\sf \:d = \frac{210}{210} = 1 \\ \\ \underline{ \boxed{ \sf \: d = 1}}$

Now we have the value of common difference also .

therefore,

$\sf \: a_5 \: = a + 4d \: \\ \\ : \implies \sf \: a_5 \: =3 + 4 \times 1 \\ \\ \sf \underline{ \boxed{ \sf a_5 \: =7}}$

Hence, required fifth term of A.P = 7 .