Question

[Sequence and Series]

for an arithmetic sequence 5th term is 17 and 17th term is 5.
then find the 22nd term of the arthematic sequence.

please help me fast it's urgent
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Answer 1

Sequence and Series

A sequence $a_1, a_2,... .. ....a_n.... ..$ is called arithmetic sequence or arithmetic progression if $a_{n+ 1} = a_n + d, n \in N$, where $a_1$ called the first term and the constant term $d$ is called the common difference of an A.P.

Let us consider an AP with first term $a$ and common difference $d$, that is, $a, a+d,a+2d....$

Then the $n^{th}$ term of the Arithmetic Sequence is $a_n = a +(n-1) d.$

Here in this question we've been given that, The 5th term of an arithmetic sequence is 17 and 17th term is 5. With this information we've been asked to find out the 22nd term of the arithmetic sequence.

Let's head to the question now:

As we are provided that, the 5th term of an arithmetic sequence is 17. So,

$\implies a+4d = 17 \qquad ----(1)$

Also we are provided that, the 17th term of an arithmetic sequence is 5. So,

$\implies a+16d = 5 \qquad ----(2)$

Subtracting the equation (1) from equation (2), we get:

$\implies (a + 16d) - (a + 4d) = 5 - 17 \\ \\ \implies a + 16d - a - 4d = 5 - 17 \\ \\ \implies a -a + 16d - 4d = -12 \\\\ \implies 16d - 4d = -12 \\\\ \implies 12d = -12 \\\\ \implies \boxed{d = -1}$

Now substituting the values of $d$ in equation (2), we get:

$\implies a + 16(-1) = 5 \\ \\ \implies a+(-16) = 5 \\ \\ \implies a - 16 = 5 \\ \\ \implies a = 5 + 16 \\ \\ \implies \boxed{a = 21}$

Now we have also the first term of an arithmetic sequence. We know that the sum of first term with 21 and the product of common difference is equal to the 22nd term. In mathematical term it would be like this;

$\implies a_{22} = a + 21d$

Substituting all the available values in it, we get:

$\implies a_{22} = 21 + 21(-1) \\ \\ \implies a_{22} = 21 + (-21) \\ \\ \implies a_{22} = 21 - 21 \\ \\ \implies \pmb{\boxed{\bm{a_{22} = 0}}}$

Therefore, the 22nd term of the arithmetic sequence is 0.

Answer 2

Answer:

$\huge \sf{\underbrace{\underline{Answer: -}}}$

Given :-

• Sequence and series (lesson name)
• For an arithmatic sequence 5th term is 17 and 17th term is 5.

□To find :-

• Find the 22nd term of an AP.

□Solution :-

• Lets first write the given data in question.
• It is given that,
• 5th term an AP is 17,
• We write it as ,

1. a+4d=17 (i)

• And 17th term of an AP is 5.
• we write it as ,

2) a+16d=5 (ii)

♧Now lets subtract the equations,

• (a+16d)-(a+4d)=5-17
• a-a+16d-4d=-12
• 12 d =-12
• d =-12/12
• d =-1

○Now,

• By substituting the value of d=-1 in equation we get that,
• a+4d=17
• a+4(-1)=17
• a-4=17
• a=17+4
• a=21.

□In the given question we should get the 22nd term of an AP So,

• a22=a+21d
• a22=21+21(-1)
• a22=21-21
• a22=0.

□Hence , the value of 22nd term of AP is 0.

□For more information :-

• https://brainly.in/question/44254825.
• search on web.

♧Hope it helps u mate .

♧Thank you .