Answer:
11. A+3-1×d+a+8-1×d=7
a+2d+a+7d=7
2a+9d=7........ (i)
&
a+7-1×d+a+14-1×d=-3
a+6d+a+13d=-3
2a+19d=-3...... (ii)
from(i) n( ii)
2a+9d=7
2a+19d=-3
------------------
-10d=10
d=-1.........
from (i) 2a+9×-1=7
2a-9=7
2a=16
a=8.....
T10=a+9d =8+9×-1
=8-9
=-1
15. Let the required term be nth term, i.e., an
Here, d = 14 – 3 = 11, a = 3
According to the Question, an = 99 + [tex]a_{25}[/tex]
∴ a + (n – 1) d = 99 + a + 24d
⇒ (n – 1) (11) = 99 + 24 (11)
(n – 1) (11) = 11 (9 + 24)
n – 1 = 33
n = 33 + 1 = 34
∴ 34th term is 99 more than its 25th term.
16. From the question it is given that,
[tex]T_9\\[/tex] =1/7
[tex]T_7[/tex] =1/9
Let us assume 'a' be the first term and 'd' be the common difference,
So, [tex]T_9[/tex] =a+8d=1/7 equation(i)
[tex]T_7[/tex] =a+6d=1/9 equation(ii)
Subtracting both equation (i) and equation (i),
(a+6d)−(a+8d)=1/9−1/7
a+6d−a−8d=(7−9)/63
−2d=−2/63
d=(−2/63)×(−1/2)
d=1/63
now, substitute the value of d in equation (ii) to find out a, we get
a+6(1/63)=1/9
a=1/9−6/63
a=(7−6)/63
a=1/63
Therefore,
[tex]T_{63}[/tex] =a+62d
=1/63+62(1/63) = 1/63+62/63
=(1+62)/63 =63/63 = 1
17. The given sequence is 3 , 7 , 11 ,15
[tex]a_2-a_1[/tex]= 7 - 3 = 4
[tex]a_3-a_2[/tex] = 11 - 7 = 4
Let 1584 be the n th term of the given sequence. then ,
[tex]a_n[/tex]= 15184
Here , a = 3 , d= 4
a + (n - 1) d = 1584
3 + (n - 1) 4 = 15184
(n- 1) = 15181/4n = 15181/4+1
n = 15185/4
But n should be a postive integer . So , 15184 is not any term of the given sequence.
18. a = 21; d = 18 - 21 = -3 .
Let -81 be the nth term.
Now,
nth term,
[tex]a_n=a+(n - 1)d[/tex]
-81 = 21 + ( n -1 ) -3
-81 = 21 -3n + 3
-81 = 24-3n
3n = 24 + 81
n = [tex]105/3[/tex]
n = 35
Therefore, the 35th term is -81.
19. Clearly, the given sequence is an A.P. with a=4; d=5
Let 124 be the term of the given sequence. Then,
[tex]a_n[/tex]=124
⟹a+(n+1)d=124
⟹4+(n−1)×5=124
⟹5n−1=124
⟹5n=125
⟹n=25
Hence, 25th term of the given sequence is 124.
20. Use formula of nth term=a+(n-1) d
68=7+(n-1) 3
61/3+1=n
here we look n is fractional but n can't be fractional so 68 is not a term of above series
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Verified answer
Answer:
11. A+3-1×d+a+8-1×d=7
a+2d+a+7d=7
2a+9d=7........ (i)
&
a+7-1×d+a+14-1×d=-3
a+6d+a+13d=-3
2a+19d=-3...... (ii)
&
from(i) n( ii)
2a+9d=7
2a+19d=-3
------------------
-10d=10
d=-1.........
from (i) 2a+9×-1=7
2a-9=7
2a=16
a=8.....
T10=a+9d =8+9×-1
=8-9
=-1
15. Let the required term be nth term, i.e., an
Here, d = 14 – 3 = 11, a = 3
According to the Question, an = 99 + [tex]a_{25}[/tex]
∴ a + (n – 1) d = 99 + a + 24d
⇒ (n – 1) (11) = 99 + 24 (11)
(n – 1) (11) = 11 (9 + 24)
n – 1 = 33
n = 33 + 1 = 34
∴ 34th term is 99 more than its 25th term.
16. From the question it is given that,
[tex]T_9\\[/tex] =1/7
[tex]T_7[/tex] =1/9
Let us assume 'a' be the first term and 'd' be the common difference,
So, [tex]T_9[/tex] =a+8d=1/7 equation(i)
[tex]T_7[/tex] =a+6d=1/9 equation(ii)
Subtracting both equation (i) and equation (i),
(a+6d)−(a+8d)=1/9−1/7
a+6d−a−8d=(7−9)/63
−2d=−2/63
d=(−2/63)×(−1/2)
d=1/63
now, substitute the value of d in equation (ii) to find out a, we get
a+6(1/63)=1/9
a=1/9−6/63
a=(7−6)/63
a=1/63
Therefore,
[tex]T_{63}[/tex] =a+62d
=1/63+62(1/63) = 1/63+62/63
=(1+62)/63 =63/63 = 1
17. The given sequence is 3 , 7 , 11 ,15
[tex]a_2-a_1[/tex]= 7 - 3 = 4
[tex]a_3-a_2[/tex] = 11 - 7 = 4
Let 1584 be the n th term of the given sequence. then ,
[tex]a_n[/tex]= 15184
Here , a = 3 , d= 4
a + (n - 1) d = 1584
3 + (n - 1) 4 = 15184
(n- 1) = 15181/4
n = 15181/4+1
n = 15185/4
But n should be a postive integer . So , 15184 is not any term of the given sequence.
18. a = 21; d = 18 - 21 = -3 .
Let -81 be the nth term.
Now,
nth term,
[tex]a_n=a+(n - 1)d[/tex]
-81 = 21 + ( n -1 ) -3
-81 = 21 -3n + 3
-81 = 24-3n
3n = 24 + 81
n = [tex]105/3[/tex]
n = 35
Therefore, the 35th term is -81.
19. Clearly, the given sequence is an A.P. with a=4; d=5
Let 124 be the term of the given sequence. Then,
[tex]a_n[/tex]=124
⟹a+(n+1)d=124
⟹4+(n−1)×5=124
⟹5n−1=124
⟹5n=125
⟹n=25
Hence, 25th term of the given sequence is 124.
20. Use formula of nth term=a+(n-1) d
68=7+(n-1) 3
61/3+1=n
here we look n is fractional but n can't be fractional so 68 is not a term of above series