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COMPETITION WINDOW
GENERAL TERM OF A G.P.
The nth terms of a G.P. is 
rth TERM FROM THE END OF A FINITE G.P.
Let a be the first term and r be the common ratio of a finite G.P. consisting of n terms, then
Also, if 
is the last term of the G.P. then,
GENERAL TERM OF A H.P.
To find the nth term of an H.P., find the nth term of the corresponding A.P. obtained by the reciprocals of the terms of the given H.P. Now the reciprocal of the nth term of an A.P., will be the nth term of the H.P.
SELECTION OF TERMS IN AN AP
Sometimes we require certain number of terms in AP. The following ways of selecting terms are generally very convenient.
It should be noted that in case of an odd number of terms, the middle term is a and the common difference is d. while in case of an even number of terms the middle terms are a – d, a + d and the common difference is 2d.
Remark-1 : If the sum of terms is not given, then select terms as a, a + d, a + 2d,....
Remark-2 : If three numbers a, b, c in order are in AP. Then
b – a = Common difference = c – b
b – a = c – b 2b = a + c
Thus, a,b,c are in AP if and only if 2b = a + c
Remark-3 :If a,b,c are in AP, then b is known as the arithmetic mean (AM) between a and c.
Remark-4 : If a, x, b are in AP Then,
2x = a + b x = 
Thus, AM between a and b is
Ex.9 The sum of three numbers in AP is –3, and their product is 8. Find the numbers.
Sol. Let the numbers be (a – d), a, (a + d). Then,
Sum = – 3 (a – d) + a + (a + d) = – 3 3a = – 3 a = – 1
Now, product = 8 (a – d) (a) (a + d) = 8 a(a2 – d2) = 8 (–1) (1 – d2) = 8 [
a = –1] d2 = 9 d = ±3
If d = 3, the numbers are –4, –1, 2. If d = –3, the numbers are 2, – 1, – 4
Thus, the numbers are –4, –1, 2 or 2, –1, –4
Ex.10 Find four numbers in AP, whose sum is 20 and the sum of whose squares is 120.
Sol. Let the numbers be (a – 3d), (a – d), (a + d), (a + 3d). Then,
Sum = 20 (a – 3d) + (a – d) + (a + d) + (a + 3d) = 20 4a = 20 a = 5
Now sum of the squares = 120
If d = 1, then the numbers are 2, 4, 6, 8. If d = –1, then the numbers are 8, 6, 4, 2.
Thus, the numbers are 2, 4, 6, 8 or 8, 6, 4, 2.
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FAQs on General Term of A H.P. and GP - Arithmetic Progressions, Class 10, Mathematics - Extra Documents, Videos & Tests for Class 10
| 1. What is the general term of an arithmetic progression (AP)? |  |
Ans. The general term of an arithmetic progression is given by the formula: $a_n = a_1 + (n-1)d$, where $a_n$ represents the nth term, $a_1$ is the first term, n is the position of the term in the sequence, and d is the common difference.
| 2. How can I find the nth term of an arithmetic progression if the first term and common difference are given? | |
Ans. To find the nth term of an arithmetic progression, you can use the formula $a_n = a_1 + (n-1)d$. Simply substitute the given values of $a_1$, n, and d into the formula and calculate the value of $a_n$.
| 3. What is the general term of a geometric progression (GP)? | |
Ans. The general term of a geometric progression is given by the formula: $a_n = a_1 \cdot r^{(n-1)}$, where $a_n$ represents the nth term, $a_1$ is the first term, n is the position of the term in the sequence, and r is the common ratio.
| 4. How can I find the nth term of a geometric progression if the first term and common ratio are given? | |
Ans. To find the nth term of a geometric progression, you can use the formula $a_n = a_1 \cdot r^{(n-1)}$. Simply substitute the given values of $a_1$, n, and r into the formula and calculate the value of $a_n$.
| 5. Can we determine the nth term of an arithmetic or geometric progression if only two terms are given? | |
Ans. No, we cannot determine the nth term of an arithmetic or geometric progression if only two terms are given. To find the nth term, we need either the common difference (for AP) or the common ratio (for GP) along with the first term. Without this information, it is not possible to find the general term of the sequence.
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