‘Progressions’ is a very easy and important topic for SSC exams. In this numbers are arranged in a series form having some logic behind them. There are three types of progressions which are AP(Arithmetic Progression), GP(Geometric Progression) and HP(Harmonic Progression).
Arithmetic Progression - In this series, the difference between the two consecutive terms will be the same. For example 1, 3, 5,7, 9. Here the difference between the two consecutive terms is 2.
Geometric Progression - In this series, the ratio of two consecutive terms will always be the same. For example 3, 9, 27,81,243. Here two consecutive terms are having the same ratios.
Harmonic Progression - Reciprocal of the terms of AP(Arithmetic Progression) is called HP(Harmonic Progression).
Arithmetic Progression :
From the below-given formula, we can obtain the nth term of the series.
$$T_{n} = a + (n-1)d$$
$$T_{n}$$ = nth term of the series
Sum of the n terms of the series = $$S{_n} = \frac{n}{2}[2a+(n-1)d]$$
a = first term of the series
d = common difference between two consecutive terms
n = number of terms
If the first and the last terms are given with the value of n, then sum of the n terms of the series = $$\frac{n}{2}\times $$ (first term + last term)
Arithmetic mean = Sum of terms/number of terms
Geometric Progression :
nth term of the series = $$a\times r^{(n-1)}$$
r = common ratio
Sum of the n terms of the series = $$S{_n} = a\times(\frac{r^{n} - 1}{r - 1})$$ when r≠1 and r>1.
Sum of the n terms of the series = $$S{_n} = a\times(\frac{1 - r^{n}}{1 - r})$$ when r≠1 and r<1.
Geometric mean = $$\sqrt[n]{a_{1}\times a_{2} . . a_{n}}$$
Harmonic Progression :
If the first two terms are P and Q, then Harmonic mean = $$\frac{2PQ}{P+Q}$$
For any two numbers, the relationship between Arithmetic mean, Geometric mean and Harmonic mean is AM≥GM≥HM.
$$GM^{2} = AM \times HM$$
Q) If the first term of AP is 97 and the common difference is 8, then find out the 10th term.
Sol.
$$T_{n} = a + (n-1)\times d$$
$$T_{10} = 97 + (10-1)\times 8$$
$$T_{10} = 97 + 9 \times 8$$
$$T_{10} = 97 + 72 = 169$$
Q) If the first term of GP is 7 and the last term of the series is 16807, then find out the sum of the GP series.
Sol.
From the given terms, we can say that the series is 7, 49, 343, 2401, 16807.
Sum of the GP series = $$a\times(\frac{r^{n}-1}{r-1})$$
= $$7\times(\frac{7^{5}-1}{7-1})$$
= $$7\times(\frac{16807-1}{6})$$
= $$7\times(\frac{16806}{6})$$
= 19607
