Question

if the 6th term of an AP is equal to four times its first term and the sum of first six terms is 75, find the first term and the common difference.​

Answer 1

Step-by-step explanation:

I hope it will help you.

Answer 2

The first term is 5 and common difference is 3

Step-by-step explanation:

Given: The 6th term of an AP is equal to four times its first term and the sum of first six terms is 75

Let a is the first term and d is the common difference and n is the number of terms

Now

$a_{6} = 4 \times a\\\\\Rightarrow a+5d=4a \\\\\Rightarrow 5d=3a\\\\\Rightarrow a=\dfrac{5d}{3} ---(i)$

$S_6=75\\\\\Rightarrow S_n=\dfrac{n}{2} (2a+(n-1)d)\\\\\Rightarrow 75=\dfrac{6}{2} (2\times \dfrac{5d}{3} +(6-1) d) ---(\text from i) \\\\\Rightarrow75=3(\dfrac{10d}{3} +5d)\\\\\Rightarrow 75=3\times \dfrac{25d}{3} \\\\\Rightarrow d= 3$

Now

$a= \dfrac{5d}{3} \\\\\Rightarrow a= \dfrac{5\times 3}{3} \\\\\Rightarrow a=5$

a= 5 and d= 3

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