Question

The sum of the first five terms of an arithmetic sequence is 150 and the sum of the first ten terms is 550.
(b) What is the eighth term?

Answer 1

sum of first five terms of AP = 150

sum of first ten terms of AP = 550

Answer 2

$Let \: 'a' \: and \: 'd'\: are \: first \: term \: and$

$common \: difference \: of \: an \: A.P$

$\pink {Sum \: of \: first \: n \: terms}$

$\pink { = \frac{n}{2} [ first \: term + last \:term ]}$

$i ) Sum \: of \: first \: five \: terms = 150$

$\implies \frac{5}{2}[ a + a + 4d ]= 150$

$\implies \frac{5}{2}[ 2a+ 4d ] = 150$

$\implies 2a+ 4d = 150 \times \frac{2}{5}$

$\implies 2a + 4d = 60 \: --(1)$

$ii) Sum \: of \: first \: ten \: terms = 550$

$\implies \frac{10}{2}[ a + a + 9d ]= 550$

$\implies 5[ 2a+ 9d ]= 550$

$\implies 2a+ 9d = \frac{550}{5}$

$\implies 2a + 9d = 110 \: --(2)$

/* Subtract equation (1) from Equation (2) , we get*/

$\implies 5d = 50$

$\implies d = \frac{50}{5}$

$\implies d = 10$

/* Put d = 10 in equation (1) , we get */

$2a + 4 \times 10 = 60$

$\implies 2a + 40 = 60$

$\implies 2a = 60 - 40$

$\implies 2a = 20$

$\implies a = \frac{20}{2}$

$\implies a = 10$

/* We know that , */

$\boxed{ \blue { n^{th} \:term (a_{n}) = a+(n-1)d }}$

$Eighth \:term = a + (8-1)d$

$= a + 7d$

$= 10 + 7 \times 10$

$= 10 + 70$

$= 80$

Therefore.,

$\red{ Eighth \:term\: of \: an \: A.P : } \green { = 80 }$

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