Question

If k+2, 2k+1 , 15 are three consecutive terms of an arithmetic sequence, then, find
a) find the value of k.
b) write the sequence.

Answer 1

Given :

• k + 2, 2k + 1, 15 are three consecutive terms of an AP

To Find :

• Value of k
• Write the sequence

Solution :

let,

• a1 = k + 2
• a2 = 2k + 1
• a3 = 15

$\underbrace{\sf{Value \: of \: k}}$

As we if the terms are consecutive then,

$\implies \sf{\dfrac{a_1 \: + \: a_3}{2} \: = \: a_2} \\ \\ \implies \sf{\dfrac{(k \: + \: 2) \: + \: 15}{2} \: = \: 2k \: + \: 1} \\ \\ \implies \sf{(k \: + \: 2) \: + \: 15 \: = \: 2(2k \: + \: 1)} \\ \\ \implies {\sf{k \: + \: 17 \: = \: 4k \: + \: 2}} \\ \\ \implies \sf{4k \: - \: k \: = \: 17 \: - \: 2} \\ \\ \implies \sf{3k \: = \: 15} \\ \\ \implies \sf{k \: = \: \dfrac{15}{3}} \\ \\ \implies \sf{k \: = \: 5}$

$\therefore$ Value of k is 5

_____________________________

$\underbrace{\sf{Sequence \: is }}$

$\sf{\bullet \: a_1 \: = \: \: k + 2 \: = \: 5 \: + \: 2 \: = \: 7} \\ \\ \sf{\bullet \: a_2 \: = \: 2k \: + \: 1 \: = 2(5) \: + \: 1 \: = \: 11} \\ \\ \sf{\bullet \: a_3 \: = \: 15}$

Answer 2

Given:-

Three consecutive terms of arithmetic sequence

k+2,2k+1,15

$\sf \bullet a_1= k+2\\ \\ \sf\bullet a_2=2k+1\\ \\ \sf\bullet a_3= 15$

To find :-

1) The value of k

2) Arithmetic sequence

Now we know that

$\boxed{\sf {a_2-a_1=a_3-a_2}}$

Put the given values

$\longmapsto\sf 2k+1-(k+2)= 15-(2k+1)\\ \\ \longmapsto\sf 2k+1-k-2= 15-2k-1\\ \\ \longmapsto\sf k-1= 14-2k\\ \\ \longmapsto\sf k+2k=14+1\\ \\ \longmapsto\sf 3k=15\\ \\ \longmapsto\sf k=\cancel{\dfrac{15}{3}}\\ \\ \longmapsto\sf k=5$

$\underline{\boxed{\textsf{\textbf{\dag \ \ k= 5}}}}$

$\underline{\underline{\sf{\bullet\ Sequence :-}}}$

$\sf \bullet a_1= k+2 [put\ the \ value \ of \ k]\\ \\ \dashrightarrow\sf a_1= 5+2= 7\\ \\ \sf\bullet a_2= 2k+1= 2\times 5+1 \\ \\ \dashrightarrow\sf a_2= 10+1=11\\ \\ \bullet\sf a_3= 15$

$\underline{\boxed{\textsf{\textbf{\dag sequence:- 7,\ 11\ , 15}}}}$